Chapter 5 - Review Exercises - Page 360: 14

$(5m+6)(2m+5)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To factor the given expression, $10m^2+37m+30 ,$ 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 $10(30)=300$ and the value of $b$ is $37 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ 12,25 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 10m^2+12m+25m+30 .\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^2+12m)+(25m+30) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 2m(5m+6)+5(5m+6) .\end{array} Factoring the $GCF= (5m+6)$ of the entire expression above results to \begin{array}{l}\require{cancel} (5m+6)(2m+5) .\end{array}

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