Rewrite \(16{m}^{2}-9{n}^{2}\) in the form \({a}^{2}-{b}^{2}\), where \(a=4m\) and \(b=3n\).
\[\begin{aligned}&\frac{4m-9n}{{(4m)}^{2}-{(3n)}^{2}}+\frac{1}{4m-3n}\\&\frac{{(m+n)}^{2}}{{m}^{2}-{n}^{2}}+\frac{{m}^{2}+mn}{{n}^{2}-mn}\end{aligned}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\begin{aligned}&\frac{4m-9n}{(4m+3n)(4m-3n)}+\frac{1}{4m-3n}\\&\frac{{(m+n)}^{2}}{{m}^{2}-{n}^{2}}+\frac{{m}^{2}+mn}{{n}^{2}-mn}\end{aligned}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\begin{aligned}&\frac{4m-9n}{(4m+3n)(4m-3n)}+\frac{1}{4m-3n}\\&\frac{{(m+n)}^{2}}{(m+n)(m-n)}+\frac{{m}^{2}+mn}{{n}^{2}-mn}\end{aligned}\]
Factor out the common term \(m\).
\[\begin{aligned}&\frac{4m-9n}{(4m+3n)(4m-3n)}+\frac{1}{4m-3n}\\&\frac{{(m+n)}^{2}}{(m+n)(m-n)}+\frac{m(m+n)}{{n}^{2}-mn}\end{aligned}\]
Factor out the common term \(n\).
\[\begin{aligned}&\frac{4m-9n}{(4m+3n)(4m-3n)}+\frac{1}{4m-3n}\\&\frac{{(m+n)}^{2}}{(m+n)(m-n)}+\frac{m(m+n)}{n(n-m)}\end{aligned}\]
Rewrite the expression with a common denominator.
\[\begin{aligned}&\frac{(4m-9n)(4m-3n)\\&{(m+n)}^{2}(m+n)(m-n)n(n-m)+(4m+3n)(4m-3n)n(n-m)+m(m+n)(4m+3n)(4m-3n)(4m-3n)\\&{(m+n)}^{2}(m+n)(m-n)}{(4m+3n)(4m-3n)(4m-3n)\\&{(m+n)}^{2}(m+n)(m-n)n(n-m)}\end{aligned}\]
Regroup terms.
\[\begin{aligned}&\frac{(4m-9n)(4m-3n)\\&{(m+n)}^{2}(m+n)(m-n)n(n-m)+n(4m+3n)(4m-3n)(n-m)+m(m+n)(4m+3n)(4m-3n)(4m-3n)\\&{(m+n)}^{2}(m+n)(m-n)}{(4m+3n)(4m-3n)(4m-3n)\\&{(m+n)}^{2}(m+n)(m-n)n(n-m)}\end{aligned}\]
((4*m-9*n)*(4*m-3*n);(m+n)^2*(m+n)*(m-n)*n*(n-m)+n*(4*m+3*n)*(4*m-3*n)*(n-m)+m*(m+n)*(4*m+3*n)*(4*m-3*n)*(4*m-3*n);(m+n)^2*(m+n)*(m-n))/((4*m+3*n)*(4*m-3*n)*(4*m-3*n);(m+n)^2*(m+n)*(m-n)*n*(n-m))
