## Intermediate Algebra: Connecting Concepts through Application

$(5m^3-7)(2m^3-3)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $10m^6-29m^3+21 ,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping. $\bf{\text{Solution Details:}}$ Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $10(21)=210$ and the value of $b$ is $-29 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -14,-15 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 10m^6-14m^3-15m^3+21 .\end{array} Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to \begin{array}{l}\require{cancel} (10m^6-14m^3)-(15m^3-21) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 2m^3(5m^3-7)-3(5m^3-7) .\end{array} Factoring the $GCF= (5m^3-7)$ of the entire expression above results to \begin{array}{l}\require{cancel} (5m^3-7)(2m^3-3) .\end{array}