Solve 3+2x/x-2-2x^3+9x+20/x^3-4x-5/x-2/x+2=

$$\frac { 3 + 2 x } { x - 2 } - \frac { 2 x ^ { 3 } + 9 x + 20 } { x ^ { 3 } - 4 x } - \frac { 5 } { x } - \frac { 2 } { x + 2 } =$$

Evaluate

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

Short Solution Steps

Factor x^{3}-4x.

$$\frac{3+2x}{x-2}-\frac{2x^{3}+9x+20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

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

$$\frac{\left(3+2x\right)x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{2x^{3}+9x+20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

Since \frac{\left(3+2x\right)x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)} and \frac{2x^{3}+9x+20}{x\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3+2x\right)x\left(x+2\right)-\left(2x^{3}+9x+20\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

Do the multiplications in \left(3+2x\right)x\left(x+2\right)-\left(2x^{3}+9x+20\right).

$$\frac{3x^{2}+6x+2x^{3}+4x^{2}-2x^{3}-9x-20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

Combine like terms in 3x^{2}+6x+2x^{3}+4x^{2}-2x^{3}-9x-20.

$$\frac{7x^{2}-3x-20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right)\left(x+2\right) and x is x\left(x-2\right)\left(x+2\right). Multiply \frac{5}{x} times \frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}.

$$\frac{7x^{2}-3x-20}{x\left(x-2\right)\left(x+2\right)}-\frac{5\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{2}{x+2}$$

Since \frac{7x^{2}-3x-20}{x\left(x-2\right)\left(x+2\right)} and \frac{5\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x^{2}-3x-20-5\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{2}{x+2}$$

Do the multiplications in 7x^{2}-3x-20-5\left(x-2\right)\left(x+2\right).

$$\frac{7x^{2}-3x-20-5x^{2}-10x+10x+20}{x\left(x-2\right)\left(x+2\right)}-\frac{2}{x+2}$$

Combine like terms in 7x^{2}-3x-20-5x^{2}-10x+10x+20.

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

Factor the expressions that are not already factored in \frac{2x^{2}-3x}{x\left(x-2\right)\left(x+2\right)}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in 2x-3-2x+4.

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

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

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

Expand

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

Short Solution Steps

Factor x^{3}-4x.

$$\frac{3+2x}{x-2}-\frac{2x^{3}+9x+20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

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

$$\frac{\left(3+2x\right)x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{2x^{3}+9x+20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

Since \frac{\left(3+2x\right)x\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)} and \frac{2x^{3}+9x+20}{x\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3+2x\right)x\left(x+2\right)-\left(2x^{3}+9x+20\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

Do the multiplications in \left(3+2x\right)x\left(x+2\right)-\left(2x^{3}+9x+20\right).

$$\frac{3x^{2}+6x+2x^{3}+4x^{2}-2x^{3}-9x-20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

Combine like terms in 3x^{2}+6x+2x^{3}+4x^{2}-2x^{3}-9x-20.

$$\frac{7x^{2}-3x-20}{x\left(x-2\right)\left(x+2\right)}-\frac{5}{x}-\frac{2}{x+2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-2\right)\left(x+2\right) and x is x\left(x-2\right)\left(x+2\right). Multiply \frac{5}{x} times \frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}.

$$\frac{7x^{2}-3x-20}{x\left(x-2\right)\left(x+2\right)}-\frac{5\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{2}{x+2}$$

Since \frac{7x^{2}-3x-20}{x\left(x-2\right)\left(x+2\right)} and \frac{5\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{7x^{2}-3x-20-5\left(x-2\right)\left(x+2\right)}{x\left(x-2\right)\left(x+2\right)}-\frac{2}{x+2}$$

Do the multiplications in 7x^{2}-3x-20-5\left(x-2\right)\left(x+2\right).

$$\frac{7x^{2}-3x-20-5x^{2}-10x+10x+20}{x\left(x-2\right)\left(x+2\right)}-\frac{2}{x+2}$$

Combine like terms in 7x^{2}-3x-20-5x^{2}-10x+10x+20.

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

Factor the expressions that are not already factored in \frac{2x^{2}-3x}{x\left(x-2\right)\left(x+2\right)}.

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

Cancel out x in both numerator and denominator.

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

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

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

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

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

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

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

Combine like terms in 2x-3-2x+4.

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

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

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