Solve x+1/x+2+x^2/x^2+8x+12-x-3/x+6

$$\frac { x + 1 } { x + 2 } + \frac { x ^ { 2 } } { x ^ { 2 } + 8 x + 12 } - \frac { x - 3 } { x + 6 }$$

Evaluate

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Short Solution Steps

Factor x^{2}+8x+12.

$$\frac{x+1}{x+2}+\frac{x^{2}}{\left(x+2\right)\left(x+6\right)}-\frac{x-3}{x+6}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and \left(x+2\right)\left(x+6\right) is \left(x+2\right)\left(x+6\right). Multiply \frac{x+1}{x+2} times \frac{x+6}{x+6}.

$$\frac{\left(x+1\right)\left(x+6\right)}{\left(x+2\right)\left(x+6\right)}+\frac{x^{2}}{\left(x+2\right)\left(x+6\right)}-\frac{x-3}{x+6}$$

Since \frac{\left(x+1\right)\left(x+6\right)}{\left(x+2\right)\left(x+6\right)} and \frac{x^{2}}{\left(x+2\right)\left(x+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)\left(x+6\right)+x^{2}}{\left(x+2\right)\left(x+6\right)}-\frac{x-3}{x+6}$$

Do the multiplications in \left(x+1\right)\left(x+6\right)+x^{2}.

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

Combine like terms in x^{2}+6x+x+6+x^{2}.

$$\frac{2x^{2}+7x+6}{\left(x+2\right)\left(x+6\right)}-\frac{x-3}{x+6}$$

Factor the expressions that are not already factored in \frac{2x^{2}+7x+6}{\left(x+2\right)\left(x+6\right)}.

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

Cancel out x+2 in both numerator and denominator.

$$\frac{2x+3}{x+6}-\frac{x-3}{x+6}$$

Since \frac{2x+3}{x+6} and \frac{x-3}{x+6} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{2x+3-x+3}{x+6}$$

Combine like terms in 2x+3-x+3.

$$\frac{x+6}{x+6}$$

Cancel out x+6 in both numerator and denominator.

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Factor

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