Solve 8/x+3+12/x+7/frac{5x+23{x^2+10x+21}}

$$\frac { \frac { 8 } { x + 3 } + \frac { 12 } { x + 7 } } { \frac { 5 x + 23 } { x ^ { 2 } + 10 x + 21 } }$$

Evaluate

$$4$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and x+7 is \left(x+3\right)\left(x+7\right). Multiply \frac{8}{x+3} times \frac{x+7}{x+7}. Multiply \frac{12}{x+7} times \frac{x+3}{x+3}.

$$\frac{\frac{8\left(x+7\right)}{\left(x+3\right)\left(x+7\right)}+\frac{12\left(x+3\right)}{\left(x+3\right)\left(x+7\right)}}{\frac{5x+23}{x^{2}+10x+21}}$$

Since \frac{8\left(x+7\right)}{\left(x+3\right)\left(x+7\right)} and \frac{12\left(x+3\right)}{\left(x+3\right)\left(x+7\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{8\left(x+7\right)+12\left(x+3\right)}{\left(x+3\right)\left(x+7\right)}}{\frac{5x+23}{x^{2}+10x+21}}$$

Do the multiplications in 8\left(x+7\right)+12\left(x+3\right).

$$\frac{\frac{8x+56+12x+36}{\left(x+3\right)\left(x+7\right)}}{\frac{5x+23}{x^{2}+10x+21}}$$

Combine like terms in 8x+56+12x+36.

$$\frac{\frac{20x+92}{\left(x+3\right)\left(x+7\right)}}{\frac{5x+23}{x^{2}+10x+21}}$$

Divide \frac{20x+92}{\left(x+3\right)\left(x+7\right)} by \frac{5x+23}{x^{2}+10x+21} by multiplying \frac{20x+92}{\left(x+3\right)\left(x+7\right)} by the reciprocal of \frac{5x+23}{x^{2}+10x+21}.

$$\frac{\left(20x+92\right)\left(x^{2}+10x+21\right)}{\left(x+3\right)\left(x+7\right)\left(5x+23\right)}$$

Factor the expressions that are not already factored.

$$\frac{4\left(x+3\right)\left(x+7\right)\left(5x+23\right)}{\left(x+3\right)\left(x+7\right)\left(5x+23\right)}$$

Cancel out \left(x+3\right)\left(x+7\right)\left(5x+23\right) in both numerator and denominator.

$$4$$

Factor

$$2^{2}$$