Solve 3/2x^4-1/x+2-x+10/2x^2-8=

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

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x^{4} and x+2 is 2\left(x+2\right)x^{4}. Multiply \frac{3}{2x^{4}} times \frac{x+2}{x+2}. Multiply \frac{1}{x+2} times \frac{2x^{4}}{2x^{4}}.

$$\frac{3\left(x+2\right)}{2\left(x+2\right)x^{4}}-\frac{2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2x^{2}-8}$$

Since \frac{3\left(x+2\right)}{2\left(x+2\right)x^{4}} and \frac{2x^{4}}{2\left(x+2\right)x^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(x+2\right)-2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2x^{2}-8}$$

Do the multiplications in 3\left(x+2\right)-2x^{4}.

$$\frac{3x+6-2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2x^{2}-8}$$

Factor 2x^{2}-8.

$$\frac{3x+6-2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2\left(x-2\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x+2\right)x^{4} and 2\left(x-2\right)\left(x+2\right) is 2\left(x-2\right)\left(x+2\right)x^{4}. Multiply \frac{3x+6-2x^{4}}{2\left(x+2\right)x^{4}} times \frac{x-2}{x-2}. Multiply \frac{x+10}{2\left(x-2\right)\left(x+2\right)} times \frac{x^{4}}{x^{4}}.

$$\frac{\left(3x+6-2x^{4}\right)\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)x^{4}}-\frac{\left(x+10\right)x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Since \frac{\left(3x+6-2x^{4}\right)\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)x^{4}} and \frac{\left(x+10\right)x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+6-2x^{4}\right)\left(x-2\right)-\left(x+10\right)x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Do the multiplications in \left(3x+6-2x^{4}\right)\left(x-2\right)-\left(x+10\right)x^{4}.

$$\frac{3x^{2}-6x+6x-12-2x^{5}+4x^{4}-x^{5}-10x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Combine like terms in 3x^{2}-6x+6x-12-2x^{5}+4x^{4}-x^{5}-10x^{4}.

$$\frac{3x^{2}-12-3x^{5}-6x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Factor the expressions that are not already factored in \frac{3x^{2}-12-3x^{5}-6x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}.

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

Cancel out x+2 in both numerator and denominator.

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

Expand 2\left(x-2\right)x^{4}.

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

Use the distributive property to multiply 3 by -x^{4}+x-2.

$$\frac{-3x^{4}+3x-6}{2x^{5}-4x^{4}}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x^{4} and x+2 is 2\left(x+2\right)x^{4}. Multiply \frac{3}{2x^{4}} times \frac{x+2}{x+2}. Multiply \frac{1}{x+2} times \frac{2x^{4}}{2x^{4}}.

$$\frac{3\left(x+2\right)}{2\left(x+2\right)x^{4}}-\frac{2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2x^{2}-8}$$

Since \frac{3\left(x+2\right)}{2\left(x+2\right)x^{4}} and \frac{2x^{4}}{2\left(x+2\right)x^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(x+2\right)-2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2x^{2}-8}$$

Do the multiplications in 3\left(x+2\right)-2x^{4}.

$$\frac{3x+6-2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2x^{2}-8}$$

Factor 2x^{2}-8.

$$\frac{3x+6-2x^{4}}{2\left(x+2\right)x^{4}}-\frac{x+10}{2\left(x-2\right)\left(x+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x+2\right)x^{4} and 2\left(x-2\right)\left(x+2\right) is 2\left(x-2\right)\left(x+2\right)x^{4}. Multiply \frac{3x+6-2x^{4}}{2\left(x+2\right)x^{4}} times \frac{x-2}{x-2}. Multiply \frac{x+10}{2\left(x-2\right)\left(x+2\right)} times \frac{x^{4}}{x^{4}}.

$$\frac{\left(3x+6-2x^{4}\right)\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)x^{4}}-\frac{\left(x+10\right)x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Since \frac{\left(3x+6-2x^{4}\right)\left(x-2\right)}{2\left(x-2\right)\left(x+2\right)x^{4}} and \frac{\left(x+10\right)x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(3x+6-2x^{4}\right)\left(x-2\right)-\left(x+10\right)x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Do the multiplications in \left(3x+6-2x^{4}\right)\left(x-2\right)-\left(x+10\right)x^{4}.

$$\frac{3x^{2}-6x+6x-12-2x^{5}+4x^{4}-x^{5}-10x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Combine like terms in 3x^{2}-6x+6x-12-2x^{5}+4x^{4}-x^{5}-10x^{4}.

$$\frac{3x^{2}-12-3x^{5}-6x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}$$

Factor the expressions that are not already factored in \frac{3x^{2}-12-3x^{5}-6x^{4}}{2\left(x-2\right)\left(x+2\right)x^{4}}.

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

Cancel out x+2 in both numerator and denominator.

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

Expand 2\left(x-2\right)x^{4}.

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

Use the distributive property to multiply 3 by -x^{4}+x-2.

$$\frac{-3x^{4}+3x-6}{2x^{5}-4x^{4}}$$