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

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

Evaluate

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

Short Solution Steps

Factor x^{2}-4. Factor x^{2}+3x+2.

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

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

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

Since \frac{x+1}{\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.

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

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

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

Combine like terms in x+1-x+2.

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

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

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

Differentiate w.r.t. x

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