Solve 5(1-2x)/(2x-1)^2+x(2x+1)(x-1)/3(x+1)

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

Evaluate

$$\frac{4x^{4}-4x^{3}-x^{2}-14x-15}{3\left(2x-1\right)\left(x+1\right)}$$

Short Solution Steps

Use the distributive property to multiply 5 by 1-2x.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+x\times \frac{\left(2x+1\right)\left(x-1\right)}{3\left(x+1\right)}$$

Use the distributive property to multiply 2x+1 by x-1 and combine like terms.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+x\times \frac{2x^{2}-x-1}{3\left(x+1\right)}$$

Use the distributive property to multiply 3 by x+1.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+x\times \frac{2x^{2}-x-1}{3x+3}$$

Express x\times \frac{2x^{2}-x-1}{3x+3} as a single fraction.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+\frac{x\left(2x^{2}-x-1\right)}{3x+3}$$

Factor 3x+3.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+\frac{x\left(2x^{2}-x-1\right)}{3\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)^{2} and 3\left(x+1\right) is 3\left(x+1\right)\left(2x-1\right)^{2}. Multiply \frac{5-10x}{\left(2x-1\right)^{2}} times \frac{3\left(x+1\right)}{3\left(x+1\right)}. Multiply \frac{x\left(2x^{2}-x-1\right)}{3\left(x+1\right)} times \frac{\left(2x-1\right)^{2}}{\left(2x-1\right)^{2}}.

$$\frac{\left(5-10x\right)\times 3\left(x+1\right)}{3\left(x+1\right)\left(2x-1\right)^{2}}+\frac{x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Since \frac{\left(5-10x\right)\times 3\left(x+1\right)}{3\left(x+1\right)\left(2x-1\right)^{2}} and \frac{x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}}{3\left(x+1\right)\left(2x-1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(5-10x\right)\times 3\left(x+1\right)+x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Do the multiplications in \left(5-10x\right)\times 3\left(x+1\right)+x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}.

$$\frac{15x+15-30x^{2}-30x+8x^{5}-8x^{4}+2x^{3}-4x^{4}+4x^{3}-x^{2}-4x^{3}+4x^{2}-x}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Combine like terms in 15x+15-30x^{2}-30x+8x^{5}-8x^{4}+2x^{3}-4x^{4}+4x^{3}-x^{2}-4x^{3}+4x^{2}-x.

$$\frac{-16x+15-27x^{2}+8x^{5}-12x^{4}+2x^{3}}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Factor the expressions that are not already factored in \frac{-16x+15-27x^{2}+8x^{5}-12x^{4}+2x^{3}}{3\left(x+1\right)\left(2x-1\right)^{2}}.

$$\frac{\left(2x-1\right)\left(4x^{4}-4x^{3}-x^{2}-14x-15\right)}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Cancel out 2x-1 in both numerator and denominator.

$$\frac{4x^{4}-4x^{3}-x^{2}-14x-15}{3\left(2x-1\right)\left(x+1\right)}$$

Expand 3\left(2x-1\right)\left(x+1\right).

$$\frac{4x^{4}-4x^{3}-x^{2}-14x-15}{6x^{2}+3x-3}$$

Expand

$$\frac{4x^{4}-4x^{3}-x^{2}-14x-15}{3\left(2x-1\right)\left(x+1\right)}$$

Short Solution Steps

Use the distributive property to multiply 5 by 1-2x.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+x\times \frac{\left(2x+1\right)\left(x-1\right)}{3\left(x+1\right)}$$

Use the distributive property to multiply 2x+1 by x-1 and combine like terms.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+x\times \frac{2x^{2}-x-1}{3\left(x+1\right)}$$

Use the distributive property to multiply 3 by x+1.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+x\times \frac{2x^{2}-x-1}{3x+3}$$

Express x\times \frac{2x^{2}-x-1}{3x+3} as a single fraction.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+\frac{x\left(2x^{2}-x-1\right)}{3x+3}$$

Factor 3x+3.

$$\frac{5-10x}{\left(2x-1\right)^{2}}+\frac{x\left(2x^{2}-x-1\right)}{3\left(x+1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2x-1\right)^{2} and 3\left(x+1\right) is 3\left(x+1\right)\left(2x-1\right)^{2}. Multiply \frac{5-10x}{\left(2x-1\right)^{2}} times \frac{3\left(x+1\right)}{3\left(x+1\right)}. Multiply \frac{x\left(2x^{2}-x-1\right)}{3\left(x+1\right)} times \frac{\left(2x-1\right)^{2}}{\left(2x-1\right)^{2}}.

$$\frac{\left(5-10x\right)\times 3\left(x+1\right)}{3\left(x+1\right)\left(2x-1\right)^{2}}+\frac{x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Since \frac{\left(5-10x\right)\times 3\left(x+1\right)}{3\left(x+1\right)\left(2x-1\right)^{2}} and \frac{x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}}{3\left(x+1\right)\left(2x-1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(5-10x\right)\times 3\left(x+1\right)+x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Do the multiplications in \left(5-10x\right)\times 3\left(x+1\right)+x\left(2x^{2}-x-1\right)\left(2x-1\right)^{2}.

$$\frac{15x+15-30x^{2}-30x+8x^{5}-8x^{4}+2x^{3}-4x^{4}+4x^{3}-x^{2}-4x^{3}+4x^{2}-x}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Combine like terms in 15x+15-30x^{2}-30x+8x^{5}-8x^{4}+2x^{3}-4x^{4}+4x^{3}-x^{2}-4x^{3}+4x^{2}-x.

$$\frac{-16x+15-27x^{2}+8x^{5}-12x^{4}+2x^{3}}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Factor the expressions that are not already factored in \frac{-16x+15-27x^{2}+8x^{5}-12x^{4}+2x^{3}}{3\left(x+1\right)\left(2x-1\right)^{2}}.

$$\frac{\left(2x-1\right)\left(4x^{4}-4x^{3}-x^{2}-14x-15\right)}{3\left(x+1\right)\left(2x-1\right)^{2}}$$

Cancel out 2x-1 in both numerator and denominator.

$$\frac{4x^{4}-4x^{3}-x^{2}-14x-15}{3\left(2x-1\right)\left(x+1\right)}$$

Expand 3\left(2x-1\right)\left(x+1\right).

$$\frac{4x^{4}-4x^{3}-x^{2}-14x-15}{6x^{2}+3x-3}$$