## Intermediate Algebra (12th Edition)

$\bf{\text{Solution Outline:}}$ To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping. $\bf{\text{Solution Details:}}$ In the given expression, $40x^2+xy+6y^2 ,$ the value of $ac$ is $40(6)=240$ and the value of $b$ is $1 .$ The possible pairs of integers whose product is $ac$ are \begin{array}{l}\require{cancel} \{1,240\}, \{2,120\}, \{3,80\}, \{4,60\}, \{5,48\}, \{6,40\}, \{8,30\}, \{10,24\}, \{12,20\}, \{15,16\}, \{-1,-240\}, \{-2,-120\}, \{-3,-80\}, \{-4,-60\}, \{-5,-48\}, \{-6,-40\}, \{-8,-30\}, \{-10,-24\}, \{-12,-20\}, \{-15,-16\} .\end{array} None of these pairs give a sum of $b$. Hence, the given expression is $\text{ not factorable with integer coefficients .}$