Chapter 5 - Polynomials and Factoring - 5.3 Factoring Trinomials of the Type ax2+bx+c - 5.3 Exercise Set - Page 327: 100

$-2(a+1)^n(a+3)^2 (a+6)$

Factoring the $GCF= (a+1)^n(a+3)^2 ,$ the given expression is equivalent to \begin{array}{l}\require{cancel} 3(a+1)^{n+1}(a+3)^2-5(a+1)^n(a+3)^3 \\\\= (a+1)^n(a+3)^2 [3(a+1)-5(a+3)] \\\\= (a+1)^n(a+3)^2 [3(a)+3(1)-5(a)-5(3)] \\\\= (a+1)^n(a+3)^2 [3a+3-5a-15] \\\\= (a+1)^n(a+3)^2 [-2a-12] \\\\= (a+1)^n(a+3)^2 (-2)(a+6) \\\\= -2(a+1)^n(a+3)^2 (a+6) .\end{array}