## Intermediate Algebra (12th Edition)

$(2t+11u)(3t-7u)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $6t^2+19tu-77u^2 ,$ use the factoring of trinomials in the form $ax^2+bx+c.$ $\bf{\text{Solution Details:}}$ In the trinomial expression above, $a= 6 ,b= 19 ,\text{ and } c= -77 .$ Using the factoring of trinomials in the form $ax^2+bx+c,$ the two numbers whose product is $ac= 6(-77)=-462$ and whose sum is $b$ are $\left\{ 33,-14 \right\}.$ Using these two numbers to decompose the middle term results to \begin{array}{l}\require{cancel} 6t^2+33tu-14tu-77u^2 .\end{array} Using factoring by grouping, the expression above is equivalent to \begin{array}{l}\require{cancel} (6t^2+33tu)-(14tu+77u^2) \\\\= 3t(2t+11u)-7u(2t+11u) \\\\= (2t+11u)(3t-7u) .\end{array}