Solve frac{6x^{2}+7x+2}{3x^2+8x+4}-x^2+2x+1/x^2-x-2+x+10/x^2-4

$$\frac { 6 x ^ { 2 } + 7 x + 2 } { 3 x ^ { 2 } + 8 x + 4 } - \frac { x ^ { 2 } + 2 x + 1 } { x ^ { 2 } - x - 2 } + \frac { x + 10 } { x ^ { 2 } - 4 }$$

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{6x^{2}+7x+2}{3x^{2}+8x+4}.

$$\frac{\left(2x+1\right)\left(3x+2\right)}{\left(x+2\right)\left(3x+2\right)}-\frac{x^{2}+2x+1}{x^{2}-x-2}+\frac{x+10}{x^{2}-4}$$

Cancel out 3x+2 in both numerator and denominator.

$$\frac{2x+1}{x+2}-\frac{x^{2}+2x+1}{x^{2}-x-2}+\frac{x+10}{x^{2}-4}$$

Factor the expressions that are not already factored in \frac{x^{2}+2x+1}{x^{2}-x-2}.

$$\frac{2x+1}{x+2}-\frac{\left(x+1\right)^{2}}{\left(x-2\right)\left(x+1\right)}+\frac{x+10}{x^{2}-4}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{2x+1}{x+2}-\frac{x+1}{x-2}+\frac{x+10}{x^{2}-4}$$

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

$$\frac{\left(2x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

Since \frac{\left(2x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x+1\right)\left(x-2\right)-\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

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

$$\frac{2x^{2}-4x+x-2-x^{2}-2x-x-2}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

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

$$\frac{x^{2}-6x-4}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

Factor x^{2}-4.

$$\frac{x^{2}-6x-4}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{\left(x-2\right)\left(x+2\right)}$$

Since \frac{x^{2}-6x-4}{\left(x-2\right)\left(x+2\right)} and \frac{x+10}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-6x-4+x+10}{\left(x-2\right)\left(x+2\right)}$$

Combine like terms in x^{2}-6x-4+x+10.

$$\frac{x^{2}-5x+6}{\left(x-2\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}-5x+6}{\left(x-2\right)\left(x+2\right)}.

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

Cancel out x-2 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{6x^{2}+7x+2}{3x^{2}+8x+4}.

$$\frac{\left(2x+1\right)\left(3x+2\right)}{\left(x+2\right)\left(3x+2\right)}-\frac{x^{2}+2x+1}{x^{2}-x-2}+\frac{x+10}{x^{2}-4}$$

Cancel out 3x+2 in both numerator and denominator.

$$\frac{2x+1}{x+2}-\frac{x^{2}+2x+1}{x^{2}-x-2}+\frac{x+10}{x^{2}-4}$$

Factor the expressions that are not already factored in \frac{x^{2}+2x+1}{x^{2}-x-2}.

$$\frac{2x+1}{x+2}-\frac{\left(x+1\right)^{2}}{\left(x-2\right)\left(x+1\right)}+\frac{x+10}{x^{2}-4}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{2x+1}{x+2}-\frac{x+1}{x-2}+\frac{x+10}{x^{2}-4}$$

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

$$\frac{\left(2x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

Since \frac{\left(2x+1\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)} and \frac{\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(2x+1\right)\left(x-2\right)-\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

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

$$\frac{2x^{2}-4x+x-2-x^{2}-2x-x-2}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

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

$$\frac{x^{2}-6x-4}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{x^{2}-4}$$

Factor x^{2}-4.

$$\frac{x^{2}-6x-4}{\left(x-2\right)\left(x+2\right)}+\frac{x+10}{\left(x-2\right)\left(x+2\right)}$$

Since \frac{x^{2}-6x-4}{\left(x-2\right)\left(x+2\right)} and \frac{x+10}{\left(x-2\right)\left(x+2\right)} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}-6x-4+x+10}{\left(x-2\right)\left(x+2\right)}$$

Combine like terms in x^{2}-6x-4+x+10.

$$\frac{x^{2}-5x+6}{\left(x-2\right)\left(x+2\right)}$$

Factor the expressions that are not already factored in \frac{x^{2}-5x+6}{\left(x-2\right)\left(x+2\right)}.

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

Cancel out x-2 in both numerator and denominator.

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