Solve {l}{2n^2(n^2+2n+1)}{-2n(n+1)(2n+1)}{+3n(n+1)}

$$\left. \begin{array} { l } { 2 n ^ { 2 } ( n ^ { 2 } + 2 n + 1 ) } \\ { - 2 n ( n + 1 ) ( 2 n + 1 ) } \\ { + 3 n ( n + 1 ) } \end{array} \right.$$

Evaluate

$$n\left(2n^{3}-n+1\right)$$

Solution Steps

Use the distributive property to multiply 2n^{2} by n^{2}+2n+1.

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

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

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

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

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

Use the distributive property to multiply -2n^{2}-2n by 2n+1 and combine like terms.

$$2n^{4}+4n^{3}+2n^{2}-4n^{3}-6n^{2}-2n+3n^{2}+3n$$

Combine 4n^{3} and -4n^{3} to get 0.

$$2n^{4}+2n^{2}-6n^{2}-2n+3n^{2}+3n$$

Combine 2n^{2} and -6n^{2} to get -4n^{2}.

$$2n^{4}-4n^{2}-2n+3n^{2}+3n$$

Combine -4n^{2} and 3n^{2} to get -n^{2}.

$$2n^{4}-n^{2}-2n+3n$$

Combine -2n and 3n to get n.

$$2n^{4}-n^{2}+n$$

Expand

$$2n^{4}-n^{2}+n$$

Solution Steps

Use the distributive property to multiply 2n^{2} by n^{2}+2n+1.

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

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

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

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

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

Use the distributive property to multiply -2n^{2}-2n by 2n+1 and combine like terms.

$$2n^{4}+4n^{3}+2n^{2}-4n^{3}-6n^{2}-2n+3n^{2}+3n$$

Combine 4n^{3} and -4n^{3} to get 0.

$$2n^{4}+2n^{2}-6n^{2}-2n+3n^{2}+3n$$

Combine 2n^{2} and -6n^{2} to get -4n^{2}.

$$2n^{4}-4n^{2}-2n+3n^{2}+3n$$

Combine -4n^{2} and 3n^{2} to get -n^{2}.

$$2n^{4}-n^{2}-2n+3n$$

Combine -2n and 3n to get n.

$$2n^{4}-n^{2}+n$$