Solve 1/x^2-x-2+3/x^2-4-1/x^2+3x+2

$$\frac { 1 } { x ^ { 2 } - x - 2 } + \frac { 3 } { x ^ { 2 } - 4 } - \frac { 1 } { x ^ { 2 } + 3 x + 2 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}-x-2. Factor x^{2}-4.

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

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

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

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

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

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

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

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

$$\frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{1}{x^{2}+3x+2}$$

Factor x^{2}+3x+2.

$$\frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}$$

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

$$\frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Since \frac{4x+5}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} and \frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4x+5-\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Do the multiplications in 4x+5-\left(x-2\right).

$$\frac{4x+5-x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}$$

Combine like terms in 4x+5-x+2.

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

Expand \left(x-2\right)\left(x+1\right)\left(x+2\right).

$$\frac{3x+7}{x^{3}+x^{2}-4x-4}$$

Differentiate w.r.t. x

$$\frac{2\left(8-7x-12x^{2}-3x^{3}\right)}{\left(\left(x+1\right)\left(x^{2}-4\right)\right)^{2}}$$