Solve 2/x^2+7x+12+4/x^2+5x+4

$$\frac { 2 } { x ^ { 2 } + 7 x + 12 } + \frac { 4 } { x ^ { 2 } + 5 x + 4 }$$

Evaluate

$$\frac{2\left(3x+7\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}$$

Short Solution Steps

Factor x^{2}+7x+12. Factor x^{2}+5x+4.

$$\frac{2}{\left(x+3\right)\left(x+4\right)}+\frac{4}{\left(x+1\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+3\right)\left(x+4\right) and \left(x+1\right)\left(x+4\right) is \left(x+1\right)\left(x+3\right)\left(x+4\right). Multiply \frac{2}{\left(x+3\right)\left(x+4\right)} times \frac{x+1}{x+1}. Multiply \frac{4}{\left(x+1\right)\left(x+4\right)} times \frac{x+3}{x+3}.

$$\frac{2\left(x+1\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}+\frac{4\left(x+3\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{2\left(x+1\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)} and \frac{4\left(x+3\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(x+1\right)+4\left(x+3\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in 2\left(x+1\right)+4\left(x+3\right).

$$\frac{2x+2+4x+12}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 2x+2+4x+12.

$$\frac{6x+14}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}$$

Expand \left(x+1\right)\left(x+3\right)\left(x+4\right).

$$\frac{6x+14}{x^{3}+8x^{2}+19x+12}$$

Factor

$$\frac{2\left(3x+7\right)}{\left(x+1\right)\left(x+3\right)\left(x+4\right)}$$