## Intermediate Algebra (12th Edition)

$(x-y+1)(x+y-1)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $x^2-y^2+2y-1 ,$ group the last $3$ terms since these form a perfect square trinomial. Then factor the trinomial. The resulting expression becomes a difference of $2$ squares. Factor this expression using the factoring of the difference of $2$ squares. $\bf{\text{Solution Details:}}$ Grouping the first $3$ terms above results to \begin{array}{l}\require{cancel} x^2-(y^2-2y+1) .\end{array} The trinomial above is a perfect square trinomial. Using $a^2\pm2ab+b^2=(a\pm b)^2,$ the expression above is equivalent to \begin{array}{l}\require{cancel} x^2-(y-1)^2 .\end{array} The expression above is a difference of $2$ squares. Using $a^2-b^2=(a+b)(a-b),$ the factored form of the expression above is \begin{array}{l}\require{cancel} [x-(y-1)][x+(y-1)] \\\\= [x-y+1][x+y-1] \\\\= (x-y+1)(x+y-1) .\end{array}