Factor \({a}^{2}+3a+2\).
Ask: Which two numbers add up to \(3\) and multiply to \(2\)?
Rewrite the expression using the above.
\[(a+1)(a+2)\]
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{{a}^{3}-8}}\times \frac{{a}^{2}+4a+3}{{a}^{2}+a-2}\]
Rewrite \({a}^{3}-8\) in the form \({a}^{3}-{b}^{3}\), where \(a=a\) and \(b=2\).
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{{a}^{3}-{2}^{3}}}\times \frac{{a}^{2}+4a+3}{{a}^{2}+a-2}\]
Use Difference of Cubes: \({a}^{3}-{b}^{3}=(a-b)({a}^{2}+ab+{b}^{2})\).
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{(a-2)({a}^{2}+(a)(2)+{2}^{2})}}\times \frac{{a}^{2}+4a+3}{{a}^{2}+a-2}\]
Simplify \({2}^{2}\) to \(4\).
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{(a-2)({a}^{2}+a\times 2+4)}}\times \frac{{a}^{2}+4a+3}{{a}^{2}+a-2}\]
Regroup terms.
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{(a-2)({a}^{2}+2a+4)}}\times \frac{{a}^{2}+4a+3}{{a}^{2}+a-2}\]
Factor \({a}^{2}+4a+3\).
Ask: Which two numbers add up to \(4\) and multiply to \(3\)?
Rewrite the expression using the above.
\[(a+1)(a+3)\]
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{(a-2)({a}^{2}+2a+4)}}\times \frac{(a+1)(a+3)}{{a}^{2}+a-2}\]
Factor \({a}^{2}+a-2\).
Ask: Which two numbers add up to \(1\) and multiply to \(-2\)?
Rewrite the expression using the above.
\[(a-1)(a+2)\]
\[\frac{\frac{(a+1)(a+2)}{{a}^{2}+2a+4}}{\frac{{(a+1)}^{2}}{(a-2)({a}^{2}+2a+4)}}\times \frac{(a+1)(a+3)}{(a-1)(a+2)}\]
Invert and multiply.
\[\frac{(a+1)(a+2)}{{a}^{2}+2a+4}\times \frac{(a-2)({a}^{2}+2a+4)}{{(a+1)}^{2}}\times \frac{(a+1)(a+3)}{(a-1)(a+2)}\]
Cancel \({a}^{2}+2a+4\).
\[(a+1)(a+2)\times \frac{a-2}{{(a+1)}^{2}}\times \frac{(a+1)(a+3)}{(a-1)(a+2)}\]
Cancel \(a+2\).
\[(a+1)\times \frac{a-2}{{(a+1)}^{2}}\times \frac{(a+1)(a+3)}{a-1}\]
Use this rule: \(\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}\).
\[\frac{(a+1)(a-2)(a+1)(a+3)}{{(a+1)}^{2}(a-1)}\]
Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[\frac{{(a+1)}^{2}(a-2)(a+3)}{{(a+1)}^{2}(a-1)}\]
Cancel \({(a+1)}^{2}\).
\[\frac{(a-2)(a+3)}{a-1}\]
((a-2)*(a+3))/(a-1)
