Solve 4/x^2+5x+6+x-2/x+3

$$\frac { 4 } { x ^ { 2 } + 5 x + 6 } + \frac { x - 2 } { x + 3 }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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