Solve left(a-2right)^2+4a/left(a+2right)^2-4a-8/frac{a^3+5a^2+4a+20{{a}^3-5{a}^2-4a+20}}

$$\frac{ \frac{ { \left(a-2 \right) }^{ 2 } +4a }{ { \left(a+2 \right) }^{ 2 } -4a-8 } }{ \frac{ { a }^{ 3 } +5 { a }^{ 2 } +4a+20 }{ { a }^{ 3 } -5 { a }^{ 2 } -4a+20 } }$$

Evaluate

$$\frac{a-5}{a+5}$$

Short Solution Steps

Divide \frac{\left(a-2\right)^{2}+4a}{\left(a+2\right)^{2}-4a-8} by \frac{a^{3}+5a^{2}+4a+20}{a^{3}-5a^{2}-4a+20} by multiplying \frac{\left(a-2\right)^{2}+4a}{\left(a+2\right)^{2}-4a-8} by the reciprocal of \frac{a^{3}+5a^{2}+4a+20}{a^{3}-5a^{2}-4a+20}.

$$\frac{\left(\left(a-2\right)^{2}+4a\right)\left(a^{3}-5a^{2}-4a+20\right)}{\left(\left(a+2\right)^{2}-4a-8\right)\left(a^{3}+5a^{2}+4a+20\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a-5\right)\left(a-2\right)\left(a+2\right)\left(a^{2}+4\right)}{\left(a-2\right)\left(a+2\right)\left(a+5\right)\left(a^{2}+4\right)}$$

Cancel out \left(a-2\right)\left(a+2\right)\left(a^{2}+4\right) in both numerator and denominator.

$$\frac{a-5}{a+5}$$

Expand

$$\frac{a-5}{a+5}$$

Short Solution Steps

Divide \frac{\left(a-2\right)^{2}+4a}{\left(a+2\right)^{2}-4a-8} by \frac{a^{3}+5a^{2}+4a+20}{a^{3}-5a^{2}-4a+20} by multiplying \frac{\left(a-2\right)^{2}+4a}{\left(a+2\right)^{2}-4a-8} by the reciprocal of \frac{a^{3}+5a^{2}+4a+20}{a^{3}-5a^{2}-4a+20}.

$$\frac{\left(\left(a-2\right)^{2}+4a\right)\left(a^{3}-5a^{2}-4a+20\right)}{\left(\left(a+2\right)^{2}-4a-8\right)\left(a^{3}+5a^{2}+4a+20\right)}$$

Factor the expressions that are not already factored.

$$\frac{\left(a-5\right)\left(a-2\right)\left(a+2\right)\left(a^{2}+4\right)}{\left(a-2\right)\left(a+2\right)\left(a+5\right)\left(a^{2}+4\right)}$$

Cancel out \left(a-2\right)\left(a+2\right)\left(a^{2}+4\right) in both numerator and denominator.

$$\frac{a-5}{a+5}$$