Solve -20/x^2+10x+24+x+16/x^2+11x+30

$$\frac { - 20 } { x ^ { 2 } + 10 x + 24 } + \frac { x + 16 } { x ^ { 2 } + 11 x + 30 }$$

Evaluate

$$\frac{x-6}{\left(x+4\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+10x+24. Factor x^{2}+11x+30.

$$\frac{-20}{\left(x+4\right)\left(x+6\right)}+\frac{x+16}{\left(x+5\right)\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+4\right)\left(x+6\right) and \left(x+5\right)\left(x+6\right) is \left(x+4\right)\left(x+5\right)\left(x+6\right). Multiply \frac{-20}{\left(x+4\right)\left(x+6\right)} times \frac{x+5}{x+5}. Multiply \frac{x+16}{\left(x+5\right)\left(x+6\right)} times \frac{x+4}{x+4}.

$$\frac{-20\left(x+5\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}+\frac{\left(x+16\right)\left(x+4\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Since \frac{-20\left(x+5\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)} and \frac{\left(x+16\right)\left(x+4\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{-20\left(x+5\right)+\left(x+16\right)\left(x+4\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Do the multiplications in -20\left(x+5\right)+\left(x+16\right)\left(x+4\right).

$$\frac{-20x-100+x^{2}+4x+16x+64}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Combine like terms in -20x-100+x^{2}+4x+16x+64.

$$\frac{-36+x^{2}}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Factor the expressions that are not already factored in \frac{-36+x^{2}}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}.

$$\frac{\left(x-6\right)\left(x+6\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{x-6}{\left(x+4\right)\left(x+5\right)}$$

Expand \left(x+4\right)\left(x+5\right).

$$\frac{x-6}{x^{2}+9x+20}$$

Expand

$$\frac{x-6}{\left(x+4\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+10x+24. Factor x^{2}+11x+30.

$$\frac{-20}{\left(x+4\right)\left(x+6\right)}+\frac{x+16}{\left(x+5\right)\left(x+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+4\right)\left(x+6\right) and \left(x+5\right)\left(x+6\right) is \left(x+4\right)\left(x+5\right)\left(x+6\right). Multiply \frac{-20}{\left(x+4\right)\left(x+6\right)} times \frac{x+5}{x+5}. Multiply \frac{x+16}{\left(x+5\right)\left(x+6\right)} times \frac{x+4}{x+4}.

$$\frac{-20\left(x+5\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}+\frac{\left(x+16\right)\left(x+4\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Since \frac{-20\left(x+5\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)} and \frac{\left(x+16\right)\left(x+4\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{-20\left(x+5\right)+\left(x+16\right)\left(x+4\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Do the multiplications in -20\left(x+5\right)+\left(x+16\right)\left(x+4\right).

$$\frac{-20x-100+x^{2}+4x+16x+64}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Combine like terms in -20x-100+x^{2}+4x+16x+64.

$$\frac{-36+x^{2}}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Factor the expressions that are not already factored in \frac{-36+x^{2}}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}.

$$\frac{\left(x-6\right)\left(x+6\right)}{\left(x+4\right)\left(x+5\right)\left(x+6\right)}$$

Cancel out x+6 in both numerator and denominator.

$$\frac{x-6}{\left(x+4\right)\left(x+5\right)}$$

Expand \left(x+4\right)\left(x+5\right).

$$\frac{x-6}{x^{2}+9x+20}$$