Solve frac{a^{2}}{a^{2}+4a+4}*frac{8a^{2}-32}{a^{3}-2a^2}+a^5-8a^2/adiv(a^2-4)

$$\frac { a ^ { 2 } } { a ^ { 2 } + 4 a + 4 } \cdot \frac { 8 a ^ { 2 } - 32 } { a ^ { 3 } - 2 a ^ { 2 } } + \frac { a ^ { 5 } - 8 a ^ { 2 } } { a } \div ( a ^ { 2 } - 4 )$$

Evaluate

$$a^{2}+4$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{8a^{2}-32}{a^{3}-2a^{2}}.

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

Cancel out a-2 in both numerator and denominator.

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

Multiply \frac{a^{2}}{a^{2}+4a+4} times \frac{8\left(a+2\right)}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a^{2} in both numerator and denominator.

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

Express \frac{\frac{a^{5}-8a^{2}}{a}}{a^{2}-4} as a single fraction.

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

Factor the expressions that are not already factored in \frac{a^{5}-8a^{2}}{a\left(a^{2}-4\right)}.

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

Cancel out a\left(a-2\right) in both numerator and denominator.

$$\frac{8\left(a+2\right)}{a^{2}+4a+4}+\frac{a\left(a^{2}+2a+4\right)}{a+2}$$

Factor a^{2}+4a+4.

$$\frac{8\left(a+2\right)}{\left(a+2\right)^{2}}+\frac{a\left(a^{2}+2a+4\right)}{a+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+2\right)^{2} and a+2 is \left(a+2\right)^{2}. Multiply \frac{a\left(a^{2}+2a+4\right)}{a+2} times \frac{a+2}{a+2}.

$$\frac{8\left(a+2\right)}{\left(a+2\right)^{2}}+\frac{a\left(a^{2}+2a+4\right)\left(a+2\right)}{\left(a+2\right)^{2}}$$

Since \frac{8\left(a+2\right)}{\left(a+2\right)^{2}} and \frac{a\left(a^{2}+2a+4\right)\left(a+2\right)}{\left(a+2\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{8\left(a+2\right)+a\left(a^{2}+2a+4\right)\left(a+2\right)}{\left(a+2\right)^{2}}$$

Do the multiplications in 8\left(a+2\right)+a\left(a^{2}+2a+4\right)\left(a+2\right).

$$\frac{8a+16+a^{4}+2a^{3}+2a^{3}+4a^{2}+4a^{2}+8a}{\left(a+2\right)^{2}}$$

Combine like terms in 8a+16+a^{4}+2a^{3}+2a^{3}+4a^{2}+4a^{2}+8a.

$$\frac{16a+16+a^{4}+4a^{3}+8a^{2}}{\left(a+2\right)^{2}}$$

Factor the expressions that are not already factored in \frac{16a+16+a^{4}+4a^{3}+8a^{2}}{\left(a+2\right)^{2}}.

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

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

$$a^{2}+4$$

Expand

$$a^{2}+4$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{8a^{2}-32}{a^{3}-2a^{2}}.

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

Cancel out a-2 in both numerator and denominator.

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

Multiply \frac{a^{2}}{a^{2}+4a+4} times \frac{8\left(a+2\right)}{a^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out a^{2} in both numerator and denominator.

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

Express \frac{\frac{a^{5}-8a^{2}}{a}}{a^{2}-4} as a single fraction.

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

Factor the expressions that are not already factored in \frac{a^{5}-8a^{2}}{a\left(a^{2}-4\right)}.

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

Cancel out a\left(a-2\right) in both numerator and denominator.

$$\frac{8\left(a+2\right)}{a^{2}+4a+4}+\frac{a\left(a^{2}+2a+4\right)}{a+2}$$

Factor a^{2}+4a+4.

$$\frac{8\left(a+2\right)}{\left(a+2\right)^{2}}+\frac{a\left(a^{2}+2a+4\right)}{a+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a+2\right)^{2} and a+2 is \left(a+2\right)^{2}. Multiply \frac{a\left(a^{2}+2a+4\right)}{a+2} times \frac{a+2}{a+2}.

$$\frac{8\left(a+2\right)}{\left(a+2\right)^{2}}+\frac{a\left(a^{2}+2a+4\right)\left(a+2\right)}{\left(a+2\right)^{2}}$$

Since \frac{8\left(a+2\right)}{\left(a+2\right)^{2}} and \frac{a\left(a^{2}+2a+4\right)\left(a+2\right)}{\left(a+2\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{8\left(a+2\right)+a\left(a^{2}+2a+4\right)\left(a+2\right)}{\left(a+2\right)^{2}}$$

Do the multiplications in 8\left(a+2\right)+a\left(a^{2}+2a+4\right)\left(a+2\right).

$$\frac{8a+16+a^{4}+2a^{3}+2a^{3}+4a^{2}+4a^{2}+8a}{\left(a+2\right)^{2}}$$

Combine like terms in 8a+16+a^{4}+2a^{3}+2a^{3}+4a^{2}+4a^{2}+8a.

$$\frac{16a+16+a^{4}+4a^{3}+8a^{2}}{\left(a+2\right)^{2}}$$

Factor the expressions that are not already factored in \frac{16a+16+a^{4}+4a^{3}+8a^{2}}{\left(a+2\right)^{2}}.

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

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

$$a^{2}+4$$