## Intermediate Algebra (12th Edition)

$(x-y+2)(x-y-2)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $x^2-2xy+y^2-4 ,$ group the first $3$ terms and factor the trinomial. Then use the factoring of the difference of $2$ squares. $\bf{\text{Solution Details:}}$ Grouping the first $3$ terms, the given expression is equivalent to \begin{array}{l}\require{cancel} (x^2-2xy+y^2)-4 .\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} (x-1y)(x-1y)-4 \\\\= (x-y)^2-4 .\end{array} The expressions $(x-y)^2$ and $4$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $(x-y)^2-4 ,$ 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} (x-y)^2-(2)^2 \\\\= [(x-y)+2][(x-y)-2] \\\\= (x-y+2)(x-y-2) .\end{array}