Intermediate Algebra (12th Edition)

$(m+n-5)(m-n+1)$
$\bf{\text{Solution Outline:}}$ Group the first $3$ terms and the last $3$ terms of the given expression, $m^2-4m+4-n^2+6n-9 ,$ and use the factoring of perfect square trinomials. Then use the factoring of the difference of $2$ squares. $\bf{\text{Solution Details:}}$ Grouping the first $3$ terms and last $3$ terms of the given expression results to \begin{array}{l}\require{cancel} (m^2-4m+4)-(n^2-6n+9) .\end{array} Using the factoring of perfect square trinomials which is given by $a^2-2ab+b^2=(a-b)^2,$ the expression above is equivalent to \begin{array}{l}\require{cancel} (m-2)^2-(n-3)^2 .\end{array} The expressions $(m-2)^2$ and $(n-3)^2$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $(m-2)^2-(n-3)^2 ,$ 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} [(m-2)+(n-3)][(m-2)-(n-3)] \\\\= [m-2+n-3][(m-2-n+3] \\\\ [m+n-5][m-n+1] \\\\= (m+n-5)(m-n+1) .\end{array}