Solve (1/2x(x+1))(1/6x(x+1)(2x+1)-2x(x+1))

$$( \frac{ 1 }{ 2 } x(x+1))( \frac{ 1 }{ 6 } x(x+1)(2x+1)-2x(x+1))$$

Evaluate

$$\frac{\left(2x-11\right)\left(x\left(x+1\right)\right)^{2}}{12}$$

Solution Steps

Use the distributive property to multiply \frac{1}{6}x by x+1.

$$\frac{1}{2}x\left(x+1\right)\left(\left(\frac{1}{6}xx+\frac{1}{6}x\right)\left(2x+1\right)-2x\left(x+1\right)\right)$$

Multiply x and x to get x^{2}.

$$\frac{1}{2}x\left(x+1\right)\left(\left(\frac{1}{6}x^{2}+\frac{1}{6}x\right)\left(2x+1\right)-2x\left(x+1\right)\right)$$

Apply the distributive property by multiplying each term of \frac{1}{6}x^{2}+\frac{1}{6}x by each term of 2x+1.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{6}x^{2}\times 2x+\frac{1}{6}x^{2}+\frac{1}{6}x\times 2x+\frac{1}{6}x-2x\left(x+1\right)\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{6}x^{3}\times 2+\frac{1}{6}x^{2}+\frac{1}{6}x\times 2x+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply x and x to get x^{2}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{6}x^{3}\times 2+\frac{1}{6}x^{2}+\frac{1}{6}x^{2}\times 2+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{6} and 2 to get \frac{2}{6}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{2}{6}x^{3}+\frac{1}{6}x^{2}+\frac{1}{6}x^{2}\times 2+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{6}x^{2}+\frac{1}{6}x^{2}\times 2+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{6} and 2 to get \frac{2}{6}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{6}x^{2}+\frac{2}{6}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{6}x^{2}+\frac{1}{3}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Combine \frac{1}{6}x^{2} and \frac{1}{3}x^{2} to get \frac{1}{2}x^{2}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\left(\frac{1}{2}xx+\frac{1}{2}x\right)\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{2}x^{2}+\frac{1}{2}x\right)\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Use the distributive property to multiply \frac{1}{2}x^{2}+\frac{1}{2}x by \frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right).

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x^{2}-2x\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Combine \frac{1}{2}x^{2} and -2x^{2} to get -\frac{3}{2}x^{2}.

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+\frac{1}{6}x-2x\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Combine \frac{1}{6}x and -2x to get -\frac{11}{6}x.

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Use the distributive property to multiply \frac{1}{2}x^{2} by \frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x.

$$\frac{1}{2}x^{2}\times \frac{1}{3}x^{3}+\frac{1}{2}x^{2}\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)x+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{1}{2}x^{5}\times \frac{1}{3}+\frac{1}{2}x^{2}\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)x+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\frac{1}{2}x^{5}\times \frac{1}{3}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)x+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}x^{5}\times \frac{1}{3}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 1}{2\times 3}x^{5}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 3}.

$$\frac{1}{6}x^{5}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{2} times -\frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}+\frac{1\left(-3\right)}{2\times 2}x^{4}+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Do the multiplications in the fraction \frac{1\left(-3\right)}{2\times 2}.

$$\frac{1}{6}x^{5}+\frac{-3}{4}x^{4}+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{2} times -\frac{11}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}+\frac{1\left(-11\right)}{2\times 6}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Do the multiplications in the fraction \frac{1\left(-11\right)}{2\times 6}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}+\frac{-11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Fraction \frac{-11}{12} can be rewritten as -\frac{11}{12} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x^{2}-2x\right)$$

Combine \frac{1}{2}x^{2} and -2x^{2} to get -\frac{3}{2}x^{2}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+\frac{1}{6}x-2x\right)$$

Combine \frac{1}{6}x and -2x to get -\frac{11}{6}x.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x\right)$$

Use the distributive property to multiply \frac{1}{2}x by \frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\times \frac{1}{3}x^{3}+\frac{1}{2}x\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x\left(-\frac{11}{6}\right)x$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x^{4}\times \frac{1}{3}+\frac{1}{2}x\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x\left(-\frac{11}{6}\right)x$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x^{4}\times \frac{1}{3}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x\left(-\frac{11}{6}\right)x$$

Multiply x and x to get x^{2}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x^{4}\times \frac{1}{3}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Multiply \frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1\times 1}{2\times 3}x^{4}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 3}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Multiply \frac{1}{2} times -\frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}+\frac{1\left(-3\right)}{2\times 2}x^{3}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Do the multiplications in the fraction \frac{1\left(-3\right)}{2\times 2}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}+\frac{-3}{4}x^{3}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Multiply \frac{1}{2} times -\frac{11}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}+\frac{1\left(-11\right)}{2\times 6}x^{2}$$

Do the multiplications in the fraction \frac{1\left(-11\right)}{2\times 6}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}+\frac{-11}{12}x^{2}$$

Fraction \frac{-11}{12} can be rewritten as -\frac{11}{12} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}-\frac{11}{12}x^{2}$$

Combine -\frac{3}{4}x^{4} and \frac{1}{6}x^{4} to get -\frac{7}{12}x^{4}.

$$\frac{1}{6}x^{5}-\frac{7}{12}x^{4}-\frac{11}{12}x^{3}-\frac{3}{4}x^{3}-\frac{11}{12}x^{2}$$

Combine -\frac{11}{12}x^{3} and -\frac{3}{4}x^{3} to get -\frac{5}{3}x^{3}.

$$\frac{1}{6}x^{5}-\frac{7}{12}x^{4}-\frac{5}{3}x^{3}-\frac{11}{12}x^{2}$$

Expand

$$\frac{x^{5}}{6}-\frac{7x^{4}}{12}-\frac{5x^{3}}{3}-\frac{11x^{2}}{12}$$

Solution Steps

Use the distributive property to multiply \frac{1}{6}x by x+1.

$$\frac{1}{2}x\left(x+1\right)\left(\left(\frac{1}{6}xx+\frac{1}{6}x\right)\left(2x+1\right)-2x\left(x+1\right)\right)$$

Multiply x and x to get x^{2}.

$$\frac{1}{2}x\left(x+1\right)\left(\left(\frac{1}{6}x^{2}+\frac{1}{6}x\right)\left(2x+1\right)-2x\left(x+1\right)\right)$$

Apply the distributive property by multiplying each term of \frac{1}{6}x^{2}+\frac{1}{6}x by each term of 2x+1.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{6}x^{2}\times 2x+\frac{1}{6}x^{2}+\frac{1}{6}x\times 2x+\frac{1}{6}x-2x\left(x+1\right)\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{6}x^{3}\times 2+\frac{1}{6}x^{2}+\frac{1}{6}x\times 2x+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply x and x to get x^{2}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{6}x^{3}\times 2+\frac{1}{6}x^{2}+\frac{1}{6}x^{2}\times 2+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{6} and 2 to get \frac{2}{6}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{2}{6}x^{3}+\frac{1}{6}x^{2}+\frac{1}{6}x^{2}\times 2+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{6}x^{2}+\frac{1}{6}x^{2}\times 2+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{6} and 2 to get \frac{2}{6}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{6}x^{2}+\frac{2}{6}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{6}x^{2}+\frac{1}{3}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Combine \frac{1}{6}x^{2} and \frac{1}{3}x^{2} to get \frac{1}{2}x^{2}.

$$\frac{1}{2}x\left(x+1\right)\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\left(\frac{1}{2}xx+\frac{1}{2}x\right)\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply x and x to get x^{2}.

$$\left(\frac{1}{2}x^{2}+\frac{1}{2}x\right)\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Use the distributive property to multiply \frac{1}{2}x^{2}+\frac{1}{2}x by \frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right).

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x^{2}-2x\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Combine \frac{1}{2}x^{2} and -2x^{2} to get -\frac{3}{2}x^{2}.

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+\frac{1}{6}x-2x\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Combine \frac{1}{6}x and -2x to get -\frac{11}{6}x.

$$\frac{1}{2}x^{2}\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Use the distributive property to multiply \frac{1}{2}x^{2} by \frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x.

$$\frac{1}{2}x^{2}\times \frac{1}{3}x^{3}+\frac{1}{2}x^{2}\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)x+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 3 to get 5.

$$\frac{1}{2}x^{5}\times \frac{1}{3}+\frac{1}{2}x^{2}\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)x+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\frac{1}{2}x^{5}\times \frac{1}{3}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)x+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{1}{2}x^{5}\times \frac{1}{3}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1\times 1}{2\times 3}x^{5}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 3}.

$$\frac{1}{6}x^{5}+\frac{1}{2}x^{4}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{2} times -\frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}+\frac{1\left(-3\right)}{2\times 2}x^{4}+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Do the multiplications in the fraction \frac{1\left(-3\right)}{2\times 2}.

$$\frac{1}{6}x^{5}+\frac{-3}{4}x^{4}+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}+\frac{1}{2}x^{3}\left(-\frac{11}{6}\right)+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Multiply \frac{1}{2} times -\frac{11}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}+\frac{1\left(-11\right)}{2\times 6}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Do the multiplications in the fraction \frac{1\left(-11\right)}{2\times 6}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}+\frac{-11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

Fraction \frac{-11}{12} can be rewritten as -\frac{11}{12} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x\left(x+1\right)\right)$$

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

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}+\frac{1}{2}x^{2}+\frac{1}{6}x-2x^{2}-2x\right)$$

Combine \frac{1}{2}x^{2} and -2x^{2} to get -\frac{3}{2}x^{2}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}+\frac{1}{6}x-2x\right)$$

Combine \frac{1}{6}x and -2x to get -\frac{11}{6}x.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\left(\frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x\right)$$

Use the distributive property to multiply \frac{1}{2}x by \frac{1}{3}x^{3}-\frac{3}{2}x^{2}-\frac{11}{6}x.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x\times \frac{1}{3}x^{3}+\frac{1}{2}x\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x\left(-\frac{11}{6}\right)x$$

To multiply powers of the same base, add their exponents. Add 1 and 3 to get 4.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x^{4}\times \frac{1}{3}+\frac{1}{2}x\left(-\frac{3}{2}\right)x^{2}+\frac{1}{2}x\left(-\frac{11}{6}\right)x$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x^{4}\times \frac{1}{3}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x\left(-\frac{11}{6}\right)x$$

Multiply x and x to get x^{2}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{2}x^{4}\times \frac{1}{3}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Multiply \frac{1}{2} times \frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1\times 1}{2\times 3}x^{4}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Do the multiplications in the fraction \frac{1\times 1}{2\times 3}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}+\frac{1}{2}x^{3}\left(-\frac{3}{2}\right)+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Multiply \frac{1}{2} times -\frac{3}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}+\frac{1\left(-3\right)}{2\times 2}x^{3}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Do the multiplications in the fraction \frac{1\left(-3\right)}{2\times 2}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}+\frac{-3}{4}x^{3}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}+\frac{1}{2}x^{2}\left(-\frac{11}{6}\right)$$

Multiply \frac{1}{2} times -\frac{11}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}+\frac{1\left(-11\right)}{2\times 6}x^{2}$$

Do the multiplications in the fraction \frac{1\left(-11\right)}{2\times 6}.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}+\frac{-11}{12}x^{2}$$

Fraction \frac{-11}{12} can be rewritten as -\frac{11}{12} by extracting the negative sign.

$$\frac{1}{6}x^{5}-\frac{3}{4}x^{4}-\frac{11}{12}x^{3}+\frac{1}{6}x^{4}-\frac{3}{4}x^{3}-\frac{11}{12}x^{2}$$

Combine -\frac{3}{4}x^{4} and \frac{1}{6}x^{4} to get -\frac{7}{12}x^{4}.

$$\frac{1}{6}x^{5}-\frac{7}{12}x^{4}-\frac{11}{12}x^{3}-\frac{3}{4}x^{3}-\frac{11}{12}x^{2}$$

Combine -\frac{11}{12}x^{3} and -\frac{3}{4}x^{3} to get -\frac{5}{3}x^{3}.

$$\frac{1}{6}x^{5}-\frac{7}{12}x^{4}-\frac{5}{3}x^{3}-\frac{11}{12}x^{2}$$