Solve 2/(x+2)-x+3/x^3+8-x^2-2x+4/x+1:3x-3/x^2+2x=

$$\frac { 2 } { ( x + 2 ) } - \frac { x + 3 } { x ^ { 3 } + 8 } - \frac { x ^ { 2 } - 2 x + 4 } { x + 1 } : \frac { 3 x - 3 } { x ^ { 2 } + 2 x } =$$

Evaluate

$$\frac{-x^{7}-10x^{4}-15x^{3}+9x^{2}-49x-15}{3\left(x^{2}-1\right)\left(x^{3}+8\right)}$$

Short Solution Steps

Divide \frac{x^{2}-2x+4}{x+1} by \frac{3x-3}{x^{2}+2x} by multiplying \frac{x^{2}-2x+4}{x+1} by the reciprocal of \frac{3x-3}{x^{2}+2x}.

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

Factor x^{3}+8.

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

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

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

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

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

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

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

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

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

Factor \left(x+1\right)\left(3x-3\right).

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x^{2}-2x+4\right) and 3\left(x-1\right)\left(x+1\right) is 3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right). Multiply \frac{2x^{2}-5x+5}{\left(x+2\right)\left(x^{2}-2x+4\right)} times \frac{3\left(x-1\right)\left(x+1\right)}{3\left(x-1\right)\left(x+1\right)}. Multiply \frac{\left(x^{2}-2x+4\right)\left(x^{2}+2x\right)}{3\left(x-1\right)\left(x+1\right)} times \frac{\left(x+2\right)\left(x^{2}-2x+4\right)}{\left(x+2\right)\left(x^{2}-2x+4\right)}.

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

Since \frac{\left(2x^{2}-5x+5\right)\times 3\left(x-1\right)\left(x+1\right)}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)} and \frac{\left(x^{2}-2x+4\right)\left(x^{2}+2x\right)\left(x+2\right)\left(x^{2}-2x+4\right)}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(2x^{2}-5x+5\right)\times 3\left(x-1\right)\left(x+1\right)-\left(x^{2}-2x+4\right)\left(x^{2}+2x\right)\left(x+2\right)\left(x^{2}-2x+4\right).

$$\frac{6x^{4}-6x^{2}-15x^{3}+15x+15x^{2}-15-x^{7}-8x^{4}-2x^{6}-16x^{3}+2x^{6}+16x^{3}+4x^{5}+32x^{2}-4x^{5}-32x^{2}-8x^{4}-64x}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)}$$

Combine like terms in 6x^{4}-6x^{2}-15x^{3}+15x+15x^{2}-15-x^{7}-8x^{4}-2x^{6}-16x^{3}+2x^{6}+16x^{3}+4x^{5}+32x^{2}-4x^{5}-32x^{2}-8x^{4}-64x.

$$\frac{-10x^{4}+9x^{2}-15x^{3}-49x-15-x^{7}}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)}$$

Expand 3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right).

$$\frac{-10x^{4}+9x^{2}-15x^{3}-49x-15-x^{7}}{3x^{5}-3x^{3}+24x^{2}-24}$$

Expand

$$-\frac{x^{7}+10x^{4}+15x^{3}-9x^{2}+49x+15}{3\left(x^{2}-1\right)\left(x^{3}+8\right)}$$

Short Solution Steps

Divide \frac{x^{2}-2x+4}{x+1} by \frac{3x-3}{x^{2}+2x} by multiplying \frac{x^{2}-2x+4}{x+1} by the reciprocal of \frac{3x-3}{x^{2}+2x}.

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

Factor x^{3}+8.

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

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

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

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

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

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

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

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

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

Factor \left(x+1\right)\left(3x-3\right).

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x+2\right)\left(x^{2}-2x+4\right) and 3\left(x-1\right)\left(x+1\right) is 3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right). Multiply \frac{2x^{2}-5x+5}{\left(x+2\right)\left(x^{2}-2x+4\right)} times \frac{3\left(x-1\right)\left(x+1\right)}{3\left(x-1\right)\left(x+1\right)}. Multiply \frac{\left(x^{2}-2x+4\right)\left(x^{2}+2x\right)}{3\left(x-1\right)\left(x+1\right)} times \frac{\left(x+2\right)\left(x^{2}-2x+4\right)}{\left(x+2\right)\left(x^{2}-2x+4\right)}.

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

Since \frac{\left(2x^{2}-5x+5\right)\times 3\left(x-1\right)\left(x+1\right)}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)} and \frac{\left(x^{2}-2x+4\right)\left(x^{2}+2x\right)\left(x+2\right)\left(x^{2}-2x+4\right)}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(2x^{2}-5x+5\right)\times 3\left(x-1\right)\left(x+1\right)-\left(x^{2}-2x+4\right)\left(x^{2}+2x\right)\left(x+2\right)\left(x^{2}-2x+4\right).

$$\frac{6x^{4}-6x^{2}-15x^{3}+15x+15x^{2}-15-x^{7}-8x^{4}-2x^{6}-16x^{3}+2x^{6}+16x^{3}+4x^{5}+32x^{2}-4x^{5}-32x^{2}-8x^{4}-64x}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)}$$

Combine like terms in 6x^{4}-6x^{2}-15x^{3}+15x+15x^{2}-15-x^{7}-8x^{4}-2x^{6}-16x^{3}+2x^{6}+16x^{3}+4x^{5}+32x^{2}-4x^{5}-32x^{2}-8x^{4}-64x.

$$\frac{-10x^{4}+9x^{2}-15x^{3}-49x-15-x^{7}}{3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right)}$$

Expand 3\left(x-1\right)\left(x+1\right)\left(x+2\right)\left(x^{2}-2x+4\right).

$$\frac{-10x^{4}+9x^{2}-15x^{3}-49x-15-x^{7}}{3x^{5}-3x^{3}+24x^{2}-24}$$