Solve 5x+5/x^2+2x-5/x^2+4x-5/x-2=

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

Evaluate

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

Short Solution Steps

Factor x^{2}+2x.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)x^{2} and x-2 is \left(x-2\right)\left(x+2\right)x^{2}. Multiply \frac{5x^{2}-10}{\left(x+2\right)x^{2}} times \frac{x-2}{x-2}. Multiply \frac{4x-5}{x-2} times \frac{\left(x+2\right)x^{2}}{\left(x+2\right)x^{2}}.

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

Since \frac{\left(5x^{2}-10\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)x^{2}} and \frac{\left(4x-5\right)\left(x+2\right)x^{2}}{\left(x-2\right)\left(x+2\right)x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(5x^{2}-10\right)\left(x-2\right)+\left(4x-5\right)\left(x+2\right)x^{2}.

$$\frac{5x^{3}-10x^{2}-10x+20+4x^{4}+8x^{3}-5x^{3}-10x^{2}}{\left(x-2\right)\left(x+2\right)x^{2}}$$

Combine like terms in 5x^{3}-10x^{2}-10x+20+4x^{4}+8x^{3}-5x^{3}-10x^{2}.

$$\frac{8x^{3}-20x^{2}-10x+20+4x^{4}}{\left(x-2\right)\left(x+2\right)x^{2}}$$

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

$$\frac{8x^{3}-20x^{2}-10x+20+4x^{4}}{x^{4}-4x^{2}}$$

Expand

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

Short Solution Steps

Factor x^{2}+2x.

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

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

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

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

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

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

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)x^{2} and x-2 is \left(x-2\right)\left(x+2\right)x^{2}. Multiply \frac{5x^{2}-10}{\left(x+2\right)x^{2}} times \frac{x-2}{x-2}. Multiply \frac{4x-5}{x-2} times \frac{\left(x+2\right)x^{2}}{\left(x+2\right)x^{2}}.

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

Since \frac{\left(5x^{2}-10\right)\left(x-2\right)}{\left(x-2\right)\left(x+2\right)x^{2}} and \frac{\left(4x-5\right)\left(x+2\right)x^{2}}{\left(x-2\right)\left(x+2\right)x^{2}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(5x^{2}-10\right)\left(x-2\right)+\left(4x-5\right)\left(x+2\right)x^{2}.

$$\frac{5x^{3}-10x^{2}-10x+20+4x^{4}+8x^{3}-5x^{3}-10x^{2}}{\left(x-2\right)\left(x+2\right)x^{2}}$$

Combine like terms in 5x^{3}-10x^{2}-10x+20+4x^{4}+8x^{3}-5x^{3}-10x^{2}.

$$\frac{8x^{3}-20x^{2}-10x+20+4x^{4}}{\left(x-2\right)\left(x+2\right)x^{2}}$$

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

$$\frac{8x^{3}-20x^{2}-10x+20+4x^{4}}{x^{4}-4x^{2}}$$