Solve frac{2x^{3}-8x^{4}+3x^{2}}{x^{2}-x+2x^3}div5x^2-20x^4/x^2+x

$$\frac { 2 x ^ { 3 } - 8 x ^ { 4 } + 3 x ^ { 2 } } { x ^ { 2 } - x + 2 x ^ { 3 } } \div \frac { 5 x ^ { 2 } - 20 x ^ { 4 } } { x ^ { 2 } + x }$$

Evaluate

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

Short Solution Steps

Divide \frac{2x^{3}-8x^{4}+3x^{2}}{x^{2}-x+2x^{3}} by \frac{5x^{2}-20x^{4}}{x^{2}+x} by multiplying \frac{2x^{3}-8x^{4}+3x^{2}}{x^{2}-x+2x^{3}} by the reciprocal of \frac{5x^{2}-20x^{4}}{x^{2}+x}.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in 1+2x.

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

Cancel out x\left(-2x-1\right)\left(x+1\right)x^{2} in both numerator and denominator.

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

Expand the expression.

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

Expand

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

Short Solution Steps

Divide \frac{2x^{3}-8x^{4}+3x^{2}}{x^{2}-x+2x^{3}} by \frac{5x^{2}-20x^{4}}{x^{2}+x} by multiplying \frac{2x^{3}-8x^{4}+3x^{2}}{x^{2}-x+2x^{3}} by the reciprocal of \frac{5x^{2}-20x^{4}}{x^{2}+x}.

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

Factor the expressions that are not already factored.

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

Extract the negative sign in 1+2x.

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

Cancel out x\left(-2x-1\right)\left(x+1\right)x^{2} in both numerator and denominator.

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

Expand the expression.

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