## Intermediate Algebra (12th Edition)

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