Chapter 5 - Polynomials and Factoring - 5.6 Factoring: A General Strategy - 5.6 Exercise Set - Page 344: 10

$(t+4)(8t-1)$

The $2$ numbers whose product is $ac= 8(-4)=-32 $ and whose sum is $b= 31 $ are ${ 32,-1 }.$ Using these $2$ numbers to decompose the middle term of the given expression, $ 8t^2+31t-4 ,$ then the factored form is \begin{array}{l} 8t^2+32t-1t-4 \\\\= (8t^2+32t)-(1t+4) \\\\= 8t(t+4)-(t+4) \\\\= (t+4)(8t-1) .\end{array}