Solve (4x^4+32x^3+34x^2/2x^3+6{x^2-8x})*(frac{{x}^2{x}^2-1}{{x}^3+64})

$$( \frac{ 4 { x }^{ 4 } +32 { x }^{ 3 } +34 { x }^{ 2 } }{ 2 { x }^{ 3 } +6 { x }^{ 2 } -8x } ) \times ( \frac{ { x }^{ 2 } { x }^{ 2 } -1 }{ { x }^{ 3 } +64 } )$$

Evaluate

$$\frac{x\left(x+1\right)\left(x^{2}+1\right)\left(2x^{2}+16x+17\right)}{\left(x+4\right)^{2}\left(x^{2}-4x+16\right)}$$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{4x^{4}+32x^{3}+34x^{2}}{2x^{3}+6x^{2}-8x}\times \frac{x^{4}-1}{x^{3}+64}$$

Factor the expressions that are not already factored in \frac{4x^{4}+32x^{3}+34x^{2}}{2x^{3}+6x^{2}-8x}.

$$\frac{2^{2}\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)x^{2}}{2x\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Cancel out 2x in both numerator and denominator.

$$\frac{2x\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

To find the opposite of -\frac{1}{2}\sqrt{30}-4, find the opposite of each term.

$$\frac{2x\left(x+\frac{1}{2}\sqrt{30}+4\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

To find the opposite of \frac{1}{2}\sqrt{30}-4, find the opposite of each term.

$$\frac{2x\left(x+\frac{1}{2}\sqrt{30}+4\right)\left(x-\frac{1}{2}\sqrt{30}+4\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Use the distributive property to multiply 2x by x+\frac{1}{2}\sqrt{30}+4.

$$\frac{\left(2x^{2}+x\sqrt{30}+8x\right)\left(x-\frac{1}{2}\sqrt{30}+4\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Use the distributive property to multiply 2x^{2}+x\sqrt{30}+8x by x-\frac{1}{2}\sqrt{30}+4 and combine like terms.

$$\frac{2x^{3}+16x^{2}-\frac{1}{2}x\left(\sqrt{30}\right)^{2}+32x}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

The square of \sqrt{30} is 30.

$$\frac{2x^{3}+16x^{2}-\frac{1}{2}x\times 30+32x}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Multiply -\frac{1}{2} and 30 to get -15.

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

Combine -15x and 32x to get 17x.

$$\frac{2x^{3}+16x^{2}+17x}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$\frac{2x^{3}+16x^{2}+17x}{x^{2}+3x-4}\times \frac{x^{4}-1}{x^{3}+64}$$

Multiply \frac{2x^{3}+16x^{2}+17x}{x^{2}+3x-4} times \frac{x^{4}-1}{x^{3}+64} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x^{3}+16x^{2}+17x\right)\left(x^{4}-1\right)}{\left(x^{2}+3x-4\right)\left(x^{3}+64\right)}$$

Factor the expressions that are not already factored.

$$\frac{2x\left(x-1\right)\left(x+1\right)\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+4\right)^{2}\left(x^{2}-4x+16\right)}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{2x\left(x+1\right)\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)\left(x^{2}+1\right)}{\left(x+4\right)^{2}\left(x^{2}-4x+16\right)}$$

Expand the expression.

$$\frac{2x^{6}+18x^{5}+35x^{4}+35x^{3}+33x^{2}+17x}{x^{4}+4x^{3}+64x+256}$$

Expand

$$\frac{2x^{6}+18x^{5}+35x^{4}+35x^{3}+33x^{2}+17x}{\left(x+4\right)\left(x^{3}+64\right)}$$

Short Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{4x^{4}+32x^{3}+34x^{2}}{2x^{3}+6x^{2}-8x}\times \frac{x^{4}-1}{x^{3}+64}$$

Factor the expressions that are not already factored in \frac{4x^{4}+32x^{3}+34x^{2}}{2x^{3}+6x^{2}-8x}.

$$\frac{2^{2}\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)x^{2}}{2x\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Cancel out 2x in both numerator and denominator.

$$\frac{2x\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

To find the opposite of -\frac{1}{2}\sqrt{30}-4, find the opposite of each term.

$$\frac{2x\left(x+\frac{1}{2}\sqrt{30}+4\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

To find the opposite of \frac{1}{2}\sqrt{30}-4, find the opposite of each term.

$$\frac{2x\left(x+\frac{1}{2}\sqrt{30}+4\right)\left(x-\frac{1}{2}\sqrt{30}+4\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Use the distributive property to multiply 2x by x+\frac{1}{2}\sqrt{30}+4.

$$\frac{\left(2x^{2}+x\sqrt{30}+8x\right)\left(x-\frac{1}{2}\sqrt{30}+4\right)}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Use the distributive property to multiply 2x^{2}+x\sqrt{30}+8x by x-\frac{1}{2}\sqrt{30}+4 and combine like terms.

$$\frac{2x^{3}+16x^{2}-\frac{1}{2}x\left(\sqrt{30}\right)^{2}+32x}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

The square of \sqrt{30} is 30.

$$\frac{2x^{3}+16x^{2}-\frac{1}{2}x\times 30+32x}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Multiply -\frac{1}{2} and 30 to get -15.

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

Combine -15x and 32x to get 17x.

$$\frac{2x^{3}+16x^{2}+17x}{\left(x-1\right)\left(x+4\right)}\times \frac{x^{4}-1}{x^{3}+64}$$

Use the distributive property to multiply x-1 by x+4 and combine like terms.

$$\frac{2x^{3}+16x^{2}+17x}{x^{2}+3x-4}\times \frac{x^{4}-1}{x^{3}+64}$$

Multiply \frac{2x^{3}+16x^{2}+17x}{x^{2}+3x-4} times \frac{x^{4}-1}{x^{3}+64} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(2x^{3}+16x^{2}+17x\right)\left(x^{4}-1\right)}{\left(x^{2}+3x-4\right)\left(x^{3}+64\right)}$$

Factor the expressions that are not already factored.

$$\frac{2x\left(x-1\right)\left(x+1\right)\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)\left(x^{2}+1\right)}{\left(x-1\right)\left(x+4\right)^{2}\left(x^{2}-4x+16\right)}$$

Cancel out x-1 in both numerator and denominator.

$$\frac{2x\left(x+1\right)\left(x-\left(-\frac{1}{2}\sqrt{30}-4\right)\right)\left(x-\left(\frac{1}{2}\sqrt{30}-4\right)\right)\left(x^{2}+1\right)}{\left(x+4\right)^{2}\left(x^{2}-4x+16\right)}$$

Expand the expression.

$$\frac{2x^{6}+18x^{5}+35x^{4}+35x^{3}+33x^{2}+17x}{x^{4}+4x^{3}+64x+256}$$