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

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

Evaluate

$$\frac{4\left(2a-1\right)}{\left(a+6\right)a^{2}}$$

Short Solution Steps

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

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10}{a+1}+\frac{\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

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.

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10+\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

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

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10-a^{2}-a-a-1}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

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

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{9-a^{2}}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Divide \frac{a^{2}-5a+6}{a^{2}+7a+6} by \frac{9-a^{2}}{a+1} by multiplying \frac{a^{2}-5a+6}{a^{2}+7a+6} by the reciprocal of \frac{9-a^{2}}{a+1}.

$$\left(\frac{\left(a^{2}-5a+6\right)\left(a+1\right)}{\left(a^{2}+7a+6\right)\left(9-a^{2}\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Factor the expressions that are not already factored in \frac{\left(a^{2}-5a+6\right)\left(a+1\right)}{\left(a^{2}+7a+6\right)\left(9-a^{2}\right)}.

$$\left(\frac{\left(a-3\right)\left(a-2\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(a+1\right)\left(a+6\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Cancel out \left(a-3\right)\left(a+1\right) in both numerator and denominator.

$$\left(\frac{a-2}{\left(-a-3\right)\left(a+6\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(-a-3\right)\left(a+6\right) and a+3 is \left(a+3\right)\left(a+6\right). Multiply \frac{a-2}{\left(-a-3\right)\left(a+6\right)} times \frac{-1}{-1}. Multiply \frac{1}{a+3} times \frac{a+6}{a+6}.

$$\left(\frac{-\left(a-2\right)}{\left(a+3\right)\left(a+6\right)}+\frac{a+6}{\left(a+3\right)\left(a+6\right)}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Since \frac{-\left(a-2\right)}{\left(a+3\right)\left(a+6\right)} and \frac{a+6}{\left(a+3\right)\left(a+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(a-2\right)+a+6}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Do the multiplications in -\left(a-2\right)+a+6.

$$\frac{-a+2+a+6}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Combine like terms in -a+2+a+6.

$$\frac{8}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Multiply \frac{8}{\left(a+3\right)\left(a+6\right)} times \frac{2a^{2}+5a-3}{2a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)\times 2a^{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{4\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)a^{2}}$$

Factor the expressions that are not already factored.

$$\frac{4\left(2a-1\right)\left(a+3\right)}{\left(a+3\right)\left(a+6\right)a^{2}}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{4\left(2a-1\right)}{\left(a+6\right)a^{2}}$$

Expand the expression.

$$\frac{8a-4}{a^{3}+6a^{2}}$$

Expand

$$\frac{4\left(2a-1\right)}{\left(a+6\right)a^{2}}$$

Short Solution Steps

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

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10}{a+1}+\frac{\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

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.

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10+\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

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

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10-a^{2}-a-a-1}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

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

$$\left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{9-a^{2}}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Divide \frac{a^{2}-5a+6}{a^{2}+7a+6} by \frac{9-a^{2}}{a+1} by multiplying \frac{a^{2}-5a+6}{a^{2}+7a+6} by the reciprocal of \frac{9-a^{2}}{a+1}.

$$\left(\frac{\left(a^{2}-5a+6\right)\left(a+1\right)}{\left(a^{2}+7a+6\right)\left(9-a^{2}\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Factor the expressions that are not already factored in \frac{\left(a^{2}-5a+6\right)\left(a+1\right)}{\left(a^{2}+7a+6\right)\left(9-a^{2}\right)}.

$$\left(\frac{\left(a-3\right)\left(a-2\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(a+1\right)\left(a+6\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Cancel out \left(a-3\right)\left(a+1\right) in both numerator and denominator.

$$\left(\frac{a-2}{\left(-a-3\right)\left(a+6\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(-a-3\right)\left(a+6\right) and a+3 is \left(a+3\right)\left(a+6\right). Multiply \frac{a-2}{\left(-a-3\right)\left(a+6\right)} times \frac{-1}{-1}. Multiply \frac{1}{a+3} times \frac{a+6}{a+6}.

$$\left(\frac{-\left(a-2\right)}{\left(a+3\right)\left(a+6\right)}+\frac{a+6}{\left(a+3\right)\left(a+6\right)}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Since \frac{-\left(a-2\right)}{\left(a+3\right)\left(a+6\right)} and \frac{a+6}{\left(a+3\right)\left(a+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(a-2\right)+a+6}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Do the multiplications in -\left(a-2\right)+a+6.

$$\frac{-a+2+a+6}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Combine like terms in -a+2+a+6.

$$\frac{8}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}$$

Multiply \frac{8}{\left(a+3\right)\left(a+6\right)} times \frac{2a^{2}+5a-3}{2a^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{8\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)\times 2a^{2}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{4\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)a^{2}}$$

Factor the expressions that are not already factored.

$$\frac{4\left(2a-1\right)\left(a+3\right)}{\left(a+3\right)\left(a+6\right)a^{2}}$$

Cancel out a+3 in both numerator and denominator.

$$\frac{4\left(2a-1\right)}{\left(a+6\right)a^{2}}$$

Expand the expression.

$$\frac{8a-4}{a^{3}+6a^{2}}$$