Chapter 5 - Polynomials and Factoring - 5.2 Factoring Trinomials of the Type x2+bx+c - 5.2 Exercise Set - Page 317: 62

$5a^6(a-5)(a+1)$

Factoring the $GCF= 5a^6 $, the given expression, $ 5a^8-20a^7-25a^6 ,$ is equivalent to \begin{array}{l} 5a^6(a^2-4a-5) .\end{array} The 2 numbers whose product is $ac= 1(-5)=-5 $ and whose sum is $b= -4 $ are $\left\{ -5,1 \right\}.$ Using these two numbers to decompose the middle term of the above trinomial, then \begin{array}{l} 5a^6(a^2-5a+1a-5) \\\\= 5a^6[(a^2-5a)+(1a-5)] \\\\= 5a^6[a(a-5)+(a-5)] \\\\= 5a^6[(a-5)(a+1)] \\\\= 5a^6(a-5)(a+1) .\end{array}