Solve (3a/a+2-a^2/a-2)div2a/a^2-4

$$( \frac { 3 a } { a + 2 } - \frac { a ^ { 2 } } { a - 2 } ) \div \frac { 2 a } { a ^ { 2 } - 4 }$$

Evaluate

$$\frac{-a^{2}+a-6}{2}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+2 and a-2 is \left(a-2\right)\left(a+2\right). Multiply \frac{3a}{a+2} times \frac{a-2}{a-2}. Multiply \frac{a^{2}}{a-2} times \frac{a+2}{a+2}.

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

Since \frac{3a\left(a-2\right)}{\left(a-2\right)\left(a+2\right)} and \frac{a^{2}\left(a+2\right)}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3a\left(a-2\right)-a^{2}\left(a+2\right).

$$\frac{\frac{3a^{2}-6a-a^{3}-2a^{2}}{\left(a-2\right)\left(a+2\right)}}{\frac{2a}{a^{2}-4}}$$

Combine like terms in 3a^{2}-6a-a^{3}-2a^{2}.

$$\frac{\frac{a^{2}-6a-a^{3}}{\left(a-2\right)\left(a+2\right)}}{\frac{2a}{a^{2}-4}}$$

Divide \frac{a^{2}-6a-a^{3}}{\left(a-2\right)\left(a+2\right)} by \frac{2a}{a^{2}-4} by multiplying \frac{a^{2}-6a-a^{3}}{\left(a-2\right)\left(a+2\right)} by the reciprocal of \frac{2a}{a^{2}-4}.

$$\frac{\left(a^{2}-6a-a^{3}\right)\left(a^{2}-4\right)}{\left(a-2\right)\left(a+2\right)\times 2a}$$

Factor the expressions that are not already factored.

$$\frac{a\left(a-2\right)\left(a+2\right)\left(-a^{2}+a-6\right)}{2a\left(a-2\right)\left(a+2\right)}$$

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

$$\frac{-a^{2}+a-6}{2}$$

Expand

$$-\frac{a^{2}}{2}+\frac{a}{2}-3$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+2 and a-2 is \left(a-2\right)\left(a+2\right). Multiply \frac{3a}{a+2} times \frac{a-2}{a-2}. Multiply \frac{a^{2}}{a-2} times \frac{a+2}{a+2}.

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

Since \frac{3a\left(a-2\right)}{\left(a-2\right)\left(a+2\right)} and \frac{a^{2}\left(a+2\right)}{\left(a-2\right)\left(a+2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3a\left(a-2\right)-a^{2}\left(a+2\right).

$$\frac{\frac{3a^{2}-6a-a^{3}-2a^{2}}{\left(a-2\right)\left(a+2\right)}}{\frac{2a}{a^{2}-4}}$$

Combine like terms in 3a^{2}-6a-a^{3}-2a^{2}.

$$\frac{\frac{a^{2}-6a-a^{3}}{\left(a-2\right)\left(a+2\right)}}{\frac{2a}{a^{2}-4}}$$

Divide \frac{a^{2}-6a-a^{3}}{\left(a-2\right)\left(a+2\right)} by \frac{2a}{a^{2}-4} by multiplying \frac{a^{2}-6a-a^{3}}{\left(a-2\right)\left(a+2\right)} by the reciprocal of \frac{2a}{a^{2}-4}.

$$\frac{\left(a^{2}-6a-a^{3}\right)\left(a^{2}-4\right)}{\left(a-2\right)\left(a+2\right)\times 2a}$$

Factor the expressions that are not already factored.

$$\frac{a\left(a-2\right)\left(a+2\right)\left(-a^{2}+a-6\right)}{2a\left(a-2\right)\left(a+2\right)}$$

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

$$\frac{-a^{2}+a-6}{2}$$