## Intermediate Algebra: Connecting Concepts through Application

$(4b^2+25c^2)(2b+5c)(2b-5c)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $16b^4-625c^4 ,$ use the factoring of the difference of $2$ squares. $\bf{\text{Solution Details:}}$ The expressions $16b^4$ and $625c^4$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $16b^4-625c^4 ,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} (4b^2)^2-(25c^2)^2 \\\\= (4b^2+25c^2)(4b^2-25c^2) .\end{array} The expressions $4b^2$ and $25c^2$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $4b^2-25c^2 ,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} (4b^2+25c^2)[(2b)^2-(5c)^2] \\\\= (4b^2+25c^2)(2b+5c)(2b-5c) .\end{array}