Solve x/x^2+5x+6+x/x^2+7x+12

$$\frac{ x }{ { x }^{ 2 } +5x+6 } + \frac{ x }{ { x }^{ 2 } +7x+12 }$$

Evaluate

$$\frac{2x}{\left(x+2\right)\left(x+4\right)}$$

Short Solution Steps

Factor x^{2}+5x+6. Factor x^{2}+7x+12.

$$\frac{x}{\left(x+2\right)\left(x+3\right)}+\frac{x}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x+3\right) and \left(x+3\right)\left(x+4\right) is \left(x+2\right)\left(x+3\right)\left(x+4\right). Multiply \frac{x}{\left(x+2\right)\left(x+3\right)} times \frac{x+4}{x+4}. Multiply \frac{x}{\left(x+3\right)\left(x+4\right)} times \frac{x+2}{x+2}.

$$\frac{x\left(x+4\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}+\frac{x\left(x+2\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{x\left(x+4\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)} and \frac{x\left(x+2\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{x\left(x+4\right)+x\left(x+2\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in x\left(x+4\right)+x\left(x+2\right).

$$\frac{x^{2}+4x+x^{2}+2x}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in x^{2}+4x+x^{2}+2x.

$$\frac{2x^{2}+6x}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{2x^{2}+6x}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}.

$$\frac{2x\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out x+3 in both numerator and denominator.

$$\frac{2x}{\left(x+2\right)\left(x+4\right)}$$

Expand \left(x+2\right)\left(x+4\right).

$$\frac{2x}{x^{2}+6x+8}$$

Factor

$$\frac{2x}{\left(x+2\right)\left(x+4\right)}$$