Solve x/x^2+5x+6-x/x^2+7x+10

$$\frac { x } { x ^ { 2 } + 5 x + 6 } - \frac { x } { x ^ { 2 } + 7 x + 10 }$$

Evaluate

$$\frac{2x}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}$$

Short Solution Steps

Factor x^{2}+5x+6. Factor x^{2}+7x+10.

$$\frac{x}{\left(x+2\right)\left(x+3\right)}-\frac{x}{\left(x+2\right)\left(x+5\right)}$$

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+2\right)\left(x+5\right) is \left(x+2\right)\left(x+3\right)\left(x+5\right). Multiply \frac{x}{\left(x+2\right)\left(x+3\right)} times \frac{x+5}{x+5}. Multiply \frac{x}{\left(x+2\right)\left(x+5\right)} times \frac{x+3}{x+3}.

$$\frac{x\left(x+5\right)}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}-\frac{x\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}$$

Since \frac{x\left(x+5\right)}{\left(x+2\right)\left(x+3\right)\left(x+5\right)} and \frac{x\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+5\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(x+5\right)-x\left(x+3\right)}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}$$

Do the multiplications in x\left(x+5\right)-x\left(x+3\right).

$$\frac{x^{2}+5x-x^{2}-3x}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}$$

Combine like terms in x^{2}+5x-x^{2}-3x.

$$\frac{2x}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}$$

Expand \left(x+2\right)\left(x+3\right)\left(x+5\right).

$$\frac{2x}{x^{3}+10x^{2}+31x+30}$$

Factor

$$\frac{2x}{\left(x+2\right)\left(x+3\right)\left(x+5\right)}$$