Otdfbt

I would use something like this

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools
begindocument
Blah blah
[f(z)=frac1382112640cdotfracAB]
where
beginalign*
 A=&,-306772802511648469920eta^4z^4+762453974480763801600eta^5z^4\
 &,-1678626210368271790080eta^5z^3-28510918043555533736160eta^4z^3\
 &,+11443138641451067779872eta^3z^3-52164076923190540413504eta^2z^2\
 &,-78145258181161076156160eta^5z^2-211306163712129371808450eta^4z^2\
 &,+228927087397104405937944eta^3z^2+999881065017543109136462eta^3z\
 &,-317254092617698017425280eta^5z-443761561344388063474665eta^4z\
 &,+82327155732241730770824eta z-514623285385260545505123eta^2z\
 &,-1010535343560043404912120eta^2-357788302700438191196160eta^5\
 &,-43808044579418934376632-214023244873618345872240eta^4\
 &,+11818373349781028079eta^3+347370177721463765064153eta
endalign*
and
[B=(417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)]
enddocument

I suggest something line the following, so the wide terms are reduced.

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools

begindocument

begingather*
beginalign*
g(eta,z)&=
parbox[t]0.85displaywidthraggedright
$-306772802511648469920eta^4z^4+
762453974480763801600eta^5z^4-
1678626210368271790080eta^5z^3-
28510918043555533736160eta^4z^3+
11443138641451067779872eta^3z^3-
52164076923190540413504eta^2z^2-
78145258181161076156160eta^5z^2-
211306163712129371808450eta^4z^2+
228927087397104405937944eta^3z^2+
999881065017543109136462eta^3z-
317254092617698017425280eta^5z-
443761561344388063474665eta^4z+
82327155732241730770824eta z-
514623285385260545505123eta^2z-
1010535343560043404912120eta^2-
357788302700438191196160eta^5-
43808044579418934376632-
214023244873618345872240eta^4+
11818373349781028079eta^3+
347370177721463765064153eta$

\[2ex]
h(z)&=(417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)
endalign*
\[2ex]
f(z)=frac1382112640fracg(eta,z)h(z)
endgather*

enddocument

Given the nature of the operations, you can probably express this in a tidy manner using matrix multiplication notation, eg:

where:

Code:

$$ f(z)=frac1382,112,640 ; fracg(eta, z)u(z) , v(z) , w(z) $$

where

$$
beginarrayll
 g(eta, z) = 
 beginbmatrix 
 beginarrayr @hspace0em r
 - & 306,772,802,511,648,469,920 \
 & 762,453,974,480,763,801,600 \
 - & 1,678,626,210,368,271,790,080 \
 - & 28,510,918,043,555,533,736,160 \
 & 11,443,138,641,451,067,779,872 \
 - & 52,164,076,923,190,540,413,504 \
 - & 78,145,258,181,161,076,156,160 \
 - & 211,306,163,712,129,371,808,450 \
 & 228,927,087,397,104,405,937,944 \
 & 999,881,065,017,543,109,136,462 \
 - & 317,254,092,617,698,017,425,280 \
 - & 443,761,561,344,388,063,474,665 \
 & 82,327,155,732,241,730,770,824 \
 - & 514,623,285,385,260,545,505,123 \
 - & 1,010,535,343,560,043,404,912,120 \
 - & 357,788,302,700,438,191,196,160 \
 - & 43,808,044,579,418,934,376,632 \
 - & 214,023,244,873,618,345,872,240 \
 & 11,818,373,349,781,028,079 \
 & 347,370,177,721,463,765,064,153
 endarray
 endbmatrix^T
 beginbmatrix
 beginarrayl
 eta^4z^4 \
 eta^5z^4 \
 eta^5z^3 \
 eta^4z^3 \
 eta^3z^3 \
 eta^2z^2 \
 eta^5z^2 \
 eta^4z^2 \
 eta^3z^2 \
 eta^3z \
 eta^5z \
 eta^4z \
 eta z \
 eta^2z \
 eta^2 \
 eta^5 \
 1 \
 eta^4 \
 eta^3 \
 eta
 endarray
 endbmatrix
 & 
 beginarrayl
 u(z) = beginbmatrix beginarrayr @hspace0em r & 417,420 \ - & 4,169,121 \ - & 15,571,312 endarrayendbmatrix^T beginbmatrix beginarrayl z^2 \ z \ 1 endarrayendbmatrix\[3em]
 v(z) = beginbmatrix beginarrayr @hspace0em r & 1,546 \ & 3,537 endarrayendbmatrix^T beginbmatrixbeginarrayl z \ 1 endarrayendbmatrix\[3em]
 w(z) = beginbmatrixbeginarrayr @hspace0em r & 3,092 \ & 17,001 endarrayendbmatrix^T beginbmatrixbeginarrayl z \ 1 endarrayendbmatrix \[3em]
 endarray
endarray
$$

PS. Having said that, given the nature of the numbers involved, I would also agree with Mefitico's point of view in the comments, i.e. it's best to create a variable with indices and express via a cleaner expression, and then refer to a table mapping those indices to the actual numbers involved.

or

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools, nccmath
begindocument
 beginmultline*medmath
f(z)=frac1382112640
 fracleft[
 beginmultlined
 -306772802511648469920eta^4z^4+762453974480763801600eta^5z^4-\
 1678626210368271790080eta^5z^3-28510918043555533736160eta^4z^3+\
 11443138641451067779872eta^3z^3-52164076923190540413504eta^2z^2-\
 78145258181161076156160eta^5z^2-211306163712129371808450eta^4z^2+\
 228927087397104405937944eta^3z^2+999881065017543109136462eta^3z-\
 317254092617698017425280eta^5z-443761561344388063474665eta^4z+\
 82327155732241730770824eta z - 514623285385260545505123eta^2z-\
 1010535343560043404912120eta^2-357788302700438191196160eta^5-\
 43808044579418934376632-214023244873618345872240eta^4+\
 11818373349781028079eta^3+347370177721463765064153eta
 endmultlinedright]
 (417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)
 endmultline*

I recommend aligning the variables and adding some form of thousand-separators, both will enhance the readability. What I also recommend (but didn't do here) is sorting by the powers of the first and then the second variable. This is a modification of JuleV's answer.

documentclassarticle
%usepackageamsmath% Loaded by mathtools
usepackagemathtools
begindocument
Blah blah
[f(z)=frac1382112640cdotfracAB]
where
[
arraycolsep=0.5pt
beginarrayrrllrll
 A=&, -306,772,802,511,648,469,920 &eta^4 &z^4 & +762,453,974,480,763,801,600 &eta^5 &z^4\
 &, -1,678,626,210,368,271,790,080 &eta^5 &z^3 & -2,8510,918,043,555,533,736,160 &eta^4 &z^3\
 &, +11,443,138,641,451,067,779,872 &eta^3 &z^3 & -5,2164,076,923,190,540,413,504 &eta^2 &z^2\
 &, -78,145,258,181,161,076,156,160 &eta^5 &z^2 & -21,1306,163,712,129,371,808,450 &eta^4 &z^2\
 &, +228,927,087,397,104,405,937,944 &eta^3 &z^2 & +99,9881,065,017,543,109,136,462 &eta^3 &z\
 &, -317,254,092,617,698,017,425,280 &eta^5 &z & -44,3761,561,344,388,063,474,665 &eta^4 &z\
 &, +82,327,155,732,241,730,770,824 &eta &z & -51,4623,285,385,260,545,505,123 &eta^2 &z\
 &,-1,010,535,343,560,043,404,912,120 &eta^2 & & -35,7788,302,700,438,191,196,160 &eta^5 &\
 &, -43,808,044,579,418,934,376,632 & & & -21,4023,244,873,618,345,872,240 &eta^4 &\
 &, +11,818,373,349,781,028,079 &eta^3 & & +34,7370,177,721,463,765,064,153 &eta &
endarray
]
and
[B=(417,420z^2-4,169,121z-15,571,312)(1,546z+3,537)(3,092z+17,001)]
enddocument

I'm sure there are also some custom packages that can do this for you but this is just using the packages you provided:

I would usually use the package breqn. That automatically line-breaks equations, and has a lot of very nice features, but uses low-level having into the maths primitives, which means it tends to make a mess of other packages what do the same thing (for example, you can't use both breqn and sansmath in the same document)

begindmath*
f(z)=frac1382112640times-306772802511648469920eta^4z^4+left(762453974480763801600eta^5z^4-1678626210368271790080eta^5z^3-28510918043555533736160eta^4z^3+11443138641451067779872eta^3z^3-52164076923190540413504eta^2z^2-78145258181161076156160eta^5z^2-211306163712129371808450eta^4z^2+228927087397104405937944eta^3z^2+999881065017543109136462eta^3z-317254092617698017425280eta^5z-443761561344388063474665eta^4z+82327155732241730770824eta z-514623285385260545505123eta^2z-1010535343560043404912120eta^2-357788302700438191196160eta^5-43808044579418934376632-214023244873618345872240eta^4+11818373349781028079eta^3+347370177721463765064153etaright)timesleft(left(417420z^2-4169121z-15571312right)left(1546z+3537right)left(3092z+17001right)right)^-1
enddmath*

which produces

IMO the right-alignment is ugly but apparently that's the AMS standard - without the brackets it left aligns all those lines like the alginat version.

Following the original disposition of the function, but using alignat, parenthesis, and fractions to emphasize its different terms.

documentclassarticle

usepackagemathtools

begindocument

beginalignat*2
& f(z) && = frac1382112640 times left( vphantomfrac1382112640 -306772802511648469920 eta^4 z^4 + 762453974480763801600 eta^5 z^4 right. \[1.5ex] 
& && -1678626210368271790080 eta^5 z^3 -28510918043555533736160 eta^4 z^3 \[1.5ex]
& && +11443138641451067779872 eta^3 z^3 -52164076923190540413504 eta^2 z^2 \[1.5ex]
& && -78145258181161076156160 eta^5 z^2 -211306163712129371808450 eta^4 z^2 \[1.5ex]
& && +228927087397104405937944 eta^3 z^2 +999881065017543109136462 eta^3 z \[1.5ex]
& && -317254092617698017425280 eta^5 z -443761561344388063474665 eta^4 z \[1.5ex]
& && +82327155732241730770824 eta z -514623285385260545505123 eta^2 z \[1.5ex]
& && -1010535343560043404912120 eta^2 -357788302700438191196160 eta^5 \[1.5ex]
& && -43808044579418934376632 -214023244873618345872240 eta^4 \[1.5ex]
& && +11818373349781028079 eta^3 +347370177721463765064153eta left. vphantomfrac1382112640 right) \[1.5ex]
& && times frac1(417420z^2-4169121z-15571312)(1546z+3537)(3092z+17001)
endalignat*

enddocument