Answer:
[tex](5d^4-8)^2[/tex]
Step-by-step explanation:
[tex]\textsf{Let }\:u=d^4[/tex]
[tex]\implies 25d^8-80d^4+64=25u^2-80u+64[/tex]
To factor a quadratic in the form [tex]ax^2+bx+c[/tex], find two numbers that multiply to [tex]ac[/tex] and sum to [tex]b[/tex] :
[tex]\implies ac=25 \cdot 64=1600[/tex]
[tex]\implies b=-80[/tex]
Two numbers that multiply to 1600 and sum to -80 are:
-40 and -40
Rewrite b as the sum of these two numbers:
[tex]\implies 25u^2-40u-40u+64[/tex]
Factor the first two terms and the last two terms separately:
[tex]\implies 5u(5u-8)-8(5u-8)[/tex]
Factor out the common term (5u - 8):
[tex]\implies (5u-8)(5u-8)[/tex]
Simplify:
[tex]\implies (5u-8)^2[/tex]
Substitute back [tex]u=d^4[/tex]
[tex]\implies (5d^4-8)^2[/tex]
Therefore:
[tex]25d^8-80d^4+64=(5d^4-8)^2[/tex]
