Rewrite \({a}^{2}-1\) in the form \({a}^{2}-{b}^{2}\), where \(a=a\) and \(b=1\).
\[\frac{{a}^{2}-{1}^{2}}{{a}^{2}-2a-15}-\frac{a+2}{a+3}\]
Use Difference of Squares: \({a}^{2}-{b}^{2}=(a+b)(a-b)\).
\[\frac{(a+1)(a-1)}{{a}^{2}-2a-15}-\frac{a+2}{a+3}\]
Factor \({a}^{2}-2a-15\).
Ask: Which two numbers add up to \(-2\) and multiply to \(-15\)?
Rewrite the expression using the above.
\[(a-5)(a+3)\]
\[\frac{(a+1)(a-1)}{(a-5)(a+3)}-\frac{a+2}{a+3}\]
Rewrite the expression with a common denominator.
\[\frac{(a+1)(a-1)-(a+2)(a-5)}{(a-5)(a+3)}\]
Expand.
\[\frac{{a}^{2}-1-{a}^{2}+5a-2a+10}{(a-5)(a+3)}\]
Collect like terms.
\[\frac{({a}^{2}-{a}^{2})+(-1+10)+(5a-2a)}{(a-5)(a+3)}\]
Simplify \(({a}^{2}-{a}^{2})+(-1+10)+(5a-2a)\) to \(9+3a\).
\[\frac{9+3a}{(a-5)(a+3)}\]
Factor out the common term \(3\).
\[\frac{3(3+a)}{(a-5)(a+3)}\]
Simplify.
\[\frac{3(3+a)}{(a-5)(3+a)}\]
Cancel \(3+a\).
\[\frac{3}{a-5}\]
3/(a-5)
