Factor out the common term \(2\).
\[\frac{\frac{2(q-1)}{{p}^{2}-25}}{\frac{4{q}^{2}-4}{p+5}}\]
Rewrite \({p}^{2}-25\) in the form \({a}^{2}-{b}^{2}\), where \(a=p\) and \(b=5\).
\[\frac{\frac{2(q-1)}{{p}^{2}-{5}^{2}}}{\frac{4{q}^{2}-4}{p+5}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{\frac{2(q-1)}{(p+5)(p-5)}}{\frac{4{q}^{2}-4}{p+5}}\]
Factor out the common term \(4\).
\[\frac{\frac{2(q-1)}{(p+5)(p-5)}}{\frac{4({q}^{2}-1)}{p+5}}\]
Rewrite \({q}^{2}-1\) in the form \({a}^{2}-{b}^{2}\), where \(a=q\) and \(b=1\).
\[\frac{\frac{2(q-1)}{(p+5)(p-5)}}{\frac{4({q}^{2}-{1}^{2})}{p+5}}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{\frac{2(q-1)}{(p+5)(p-5)}}{\frac{4(q+1)(q-1)}{p+5}}\]
Invert and multiply.
\[\frac{2(q-1)}{(p+5)(p-5)}\times \frac{p+5}{4(q+1)(q-1)}\]
Cancel \(p+5\).
\[\frac{2(q-1)}{p-5}\times \frac{1}{4(q+1)(q-1)}\]
Cancel \(q-1\).
\[\frac{2}{p-5}\times \frac{1}{4(q+1)}\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{2\times 1}{(p-5)\times 4(q+1)}\]
Simplify \(2\times 1\) to \(2\).
\[\frac{2}{(p-5)\times 4(q+1)}\]
Regroup terms.
\[\frac{2}{4(p-5)(q+1)}\]
Simplify.
\[\frac{1}{2(p-5)(q+1)}\]
1/(2*(p-5)*(q+1))
