## Intermediate Algebra (12th Edition)

$(3+a-b)(3-a+b)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $9-a^2+2ab-b^2 ,$ group the last $3$ terms and factor the trinomial. Then use the factoring of the difference of $2$ squares. $\bf{\text{Solution Details:}}$ Grouping the last $3$ terms, the given expression is equivalent to \begin{array}{l}\require{cancel} 9-(a^2-2ab+b^2) .\end{array} In the trinomial expression above, $b= -2 ,\text{ and } c= 1 .$ Using the factoring of trinomials in the form $x^2+bx+c,$ the two numbers whose product is $c$ and whose sum is $b$ are $\left\{ -1,-1 \right\}.$ Using these two numbers, the factored form of the expression above is \begin{array}{l}\require{cancel} 9-(a-b)(a-b) \\\\ 9-(a-b)^2 .\end{array} The expressions $9$ and $(a-b)^2$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $9-(a-b)^2 ,$ 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} (3)^2-(a-b)^2 \\\\= [3+(a-b)][3-(a-b)] \\\\= (3+a-b)(3-a+b) .\end{array}