Solve (a+3/a^2-4+a/6-a-a^2)div4a+9/5a-10

$$( \frac { a + 3 } { a ^ { 2 } - 4 } + \frac { a } { 6 - a - a ^ { 2 } } ) \div \frac { 4 a + 9 } { 5 a - 10 }$$

Evaluate

$$\frac{5}{\left(a+2\right)\left(a+3\right)}$$

Short Solution Steps

Factor a^{2}-4. Factor 6-a-a^{2}.

$$\frac{\frac{a+3}{\left(a-2\right)\left(a+2\right)}+\frac{a}{\left(a+3\right)\left(-a+2\right)}}{\frac{4a+9}{5a-10}}$$

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

$$\frac{\frac{\left(a+3\right)\left(a+3\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}+\frac{a\left(-1\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Since \frac{\left(a+3\right)\left(a+3\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} and \frac{a\left(-1\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(a+3\right)\left(a+3\right)+a\left(-1\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Do the multiplications in \left(a+3\right)\left(a+3\right)+a\left(-1\right)\left(a+2\right).

$$\frac{\frac{a^{2}+3a+3a+9-a^{2}-2a}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Combine like terms in a^{2}+3a+3a+9-a^{2}-2a.

$$\frac{\frac{4a+9}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Divide \frac{4a+9}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} by \frac{4a+9}{5a-10} by multiplying \frac{4a+9}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} by the reciprocal of \frac{4a+9}{5a-10}.

$$\frac{\left(4a+9\right)\left(5a-10\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)\left(4a+9\right)}$$

Cancel out 4a+9 in both numerator and denominator.

$$\frac{5a-10}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Factor the expressions that are not already factored.

$$\frac{5\left(a-2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Cancel out a-2 in both numerator and denominator.

$$\frac{5}{\left(a+2\right)\left(a+3\right)}$$

Expand the expression.

$$\frac{5}{a^{2}+5a+6}$$

Expand

$$\frac{5}{\left(a+2\right)\left(a+3\right)}$$

Short Solution Steps

Factor a^{2}-4. Factor 6-a-a^{2}.

$$\frac{\frac{a+3}{\left(a-2\right)\left(a+2\right)}+\frac{a}{\left(a+3\right)\left(-a+2\right)}}{\frac{4a+9}{5a-10}}$$

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

$$\frac{\frac{\left(a+3\right)\left(a+3\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}+\frac{a\left(-1\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Since \frac{\left(a+3\right)\left(a+3\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} and \frac{a\left(-1\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(a+3\right)\left(a+3\right)+a\left(-1\right)\left(a+2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Do the multiplications in \left(a+3\right)\left(a+3\right)+a\left(-1\right)\left(a+2\right).

$$\frac{\frac{a^{2}+3a+3a+9-a^{2}-2a}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Combine like terms in a^{2}+3a+3a+9-a^{2}-2a.

$$\frac{\frac{4a+9}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}}{\frac{4a+9}{5a-10}}$$

Divide \frac{4a+9}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} by \frac{4a+9}{5a-10} by multiplying \frac{4a+9}{\left(a-2\right)\left(a+2\right)\left(a+3\right)} by the reciprocal of \frac{4a+9}{5a-10}.

$$\frac{\left(4a+9\right)\left(5a-10\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)\left(4a+9\right)}$$

Cancel out 4a+9 in both numerator and denominator.

$$\frac{5a-10}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Factor the expressions that are not already factored.

$$\frac{5\left(a-2\right)}{\left(a-2\right)\left(a+2\right)\left(a+3\right)}$$

Cancel out a-2 in both numerator and denominator.

$$\frac{5}{\left(a+2\right)\left(a+3\right)}$$

Expand the expression.

$$\frac{5}{a^{2}+5a+6}$$