Solve 2-5a+6/2+7a+6:(2a+10/a+1-a-1)+1/a+3]*2a^2+5a-3/2a^2

$$\frac { 2 - 5 a + 6 } { 2 + 7 a + 6 } : ( \frac { 2 a + 10 } { a + 1 } - a - 1 ) + \frac { 1 } { a + 3 } ] \cdot \frac { 2 a ^ { 2 } + 5 a - 3 } { 2 a ^ { 2 } }$$

Evaluate

$$\frac{4\left(3a^{2}-4a-8\right)}{\left(7a+8\right)\left(a^{2}-9\right)}$$

Short Solution Steps

Add 2 and 6 to get 8.

$$\frac{\frac{8-5a}{2+7a+6}}{\frac{2a+10}{a+1}-a-1}+\frac{1}{a+3}$$

Add 2 and 6 to get 8.

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10}{a+1}-a-1}+\frac{1}{a+3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-1 times \frac{a+1}{a+1}.

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10}{a+1}+\frac{\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}$$

Since \frac{2a+10}{a+1} and \frac{\left(-a-1\right)\left(a+1\right)}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10+\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}$$

Do the multiplications in 2a+10+\left(-a-1\right)\left(a+1\right).

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10-a^{2}-a-a-1}{a+1}}+\frac{1}{a+3}$$

Combine like terms in 2a+10-a^{2}-a-a-1.

$$\frac{\frac{8-5a}{8+7a}}{\frac{9-a^{2}}{a+1}}+\frac{1}{a+3}$$

Divide \frac{8-5a}{8+7a} by \frac{9-a^{2}}{a+1} by multiplying \frac{8-5a}{8+7a} by the reciprocal of \frac{9-a^{2}}{a+1}.

$$\frac{\left(8-5a\right)\left(a+1\right)}{\left(8+7a\right)\left(9-a^{2}\right)}+\frac{1}{a+3}$$

Factor \left(8+7a\right)\left(9-a^{2}\right).

$$\frac{\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(7a+8\right)}+\frac{1}{a+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-3\right)\left(-a-3\right)\left(7a+8\right) and a+3 is \left(a-3\right)\left(a+3\right)\left(7a+8\right). Multiply \frac{\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(7a+8\right)} times \frac{-1}{-1}. Multiply \frac{1}{a+3} times \frac{\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(7a+8\right)}.

$$\frac{-\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}+\frac{\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Since \frac{-\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)} and \frac{\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(8-5a\right)\left(a+1\right)+\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Do the multiplications in -\left(8-5a\right)\left(a+1\right)+\left(a-3\right)\left(7a+8\right).

$$\frac{-8a-8+5a^{2}+5a+7a^{2}+8a-21a-24}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Combine like terms in -8a-8+5a^{2}+5a+7a^{2}+8a-21a-24.

$$\frac{-16a-32+12a^{2}}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Expand \left(a-3\right)\left(a+3\right)\left(7a+8\right).

$$\frac{-16a-32+12a^{2}}{7a^{3}+8a^{2}-63a-72}$$

Expand

$$-\frac{4\left(3a^{2}-4a-8\right)}{\left(7a+8\right)\left(9-a^{2}\right)}$$

Short Solution Steps

Add 2 and 6 to get 8.

$$\frac{\frac{8-5a}{2+7a+6}}{\frac{2a+10}{a+1}-a-1}+\frac{1}{a+3}$$

Add 2 and 6 to get 8.

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10}{a+1}-a-1}+\frac{1}{a+3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -a-1 times \frac{a+1}{a+1}.

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10}{a+1}+\frac{\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}$$

Since \frac{2a+10}{a+1} and \frac{\left(-a-1\right)\left(a+1\right)}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10+\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}$$

Do the multiplications in 2a+10+\left(-a-1\right)\left(a+1\right).

$$\frac{\frac{8-5a}{8+7a}}{\frac{2a+10-a^{2}-a-a-1}{a+1}}+\frac{1}{a+3}$$

Combine like terms in 2a+10-a^{2}-a-a-1.

$$\frac{\frac{8-5a}{8+7a}}{\frac{9-a^{2}}{a+1}}+\frac{1}{a+3}$$

Divide \frac{8-5a}{8+7a} by \frac{9-a^{2}}{a+1} by multiplying \frac{8-5a}{8+7a} by the reciprocal of \frac{9-a^{2}}{a+1}.

$$\frac{\left(8-5a\right)\left(a+1\right)}{\left(8+7a\right)\left(9-a^{2}\right)}+\frac{1}{a+3}$$

Factor \left(8+7a\right)\left(9-a^{2}\right).

$$\frac{\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(7a+8\right)}+\frac{1}{a+3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-3\right)\left(-a-3\right)\left(7a+8\right) and a+3 is \left(a-3\right)\left(a+3\right)\left(7a+8\right). Multiply \frac{\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(7a+8\right)} times \frac{-1}{-1}. Multiply \frac{1}{a+3} times \frac{\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(7a+8\right)}.

$$\frac{-\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}+\frac{\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Since \frac{-\left(8-5a\right)\left(a+1\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)} and \frac{\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(8-5a\right)\left(a+1\right)+\left(a-3\right)\left(7a+8\right)}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Do the multiplications in -\left(8-5a\right)\left(a+1\right)+\left(a-3\right)\left(7a+8\right).

$$\frac{-8a-8+5a^{2}+5a+7a^{2}+8a-21a-24}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Combine like terms in -8a-8+5a^{2}+5a+7a^{2}+8a-21a-24.

$$\frac{-16a-32+12a^{2}}{\left(a-3\right)\left(a+3\right)\left(7a+8\right)}$$

Expand \left(a-3\right)\left(a+3\right)\left(7a+8\right).

$$\frac{-16a-32+12a^{2}}{7a^{3}+8a^{2}-63a-72}$$