Solve (m+2n)(m-2n)^2+2mn(m+2n)+(m-2n)(m^2+4n^2+2mn)

$$( m + 2 n ) ( m - 2 n ) ^ { 2 } + 2 m n ( m + 2 n ) + ( m - 2 n ) ( m ^ { 2 } + 4 n ^ { 2 } + 2 m n )$$

Evaluate

$$2m^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-2n\right)^{2}.

$$\left(m+2n\right)\left(m^{2}-4mn+4n^{2}\right)+2mn\left(m+2n\right)+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Use the distributive property to multiply m+2n by m^{2}-4mn+4n^{2} and combine like terms.

$$m^{3}-2nm^{2}-4mn^{2}+8n^{3}+2mn\left(m+2n\right)+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Use the distributive property to multiply 2mn by m+2n.

$$m^{3}-2nm^{2}-4mn^{2}+8n^{3}+2nm^{2}+4mn^{2}+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Combine -2nm^{2} and 2nm^{2} to get 0.

$$m^{3}-4mn^{2}+8n^{3}+4mn^{2}+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Combine -4mn^{2} and 4mn^{2} to get 0.

$$m^{3}+8n^{3}+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Use the distributive property to multiply m-2n by m^{2}+4n^{2}+2mn and combine like terms.

$$m^{3}+8n^{3}+m^{3}-8n^{3}$$

Combine m^{3} and m^{3} to get 2m^{3}.

$$2m^{3}+8n^{3}-8n^{3}$$

Combine 8n^{3} and -8n^{3} to get 0.

$$2m^{3}$$

Expand

$$2m^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(m-2n\right)^{2}.

$$\left(m+2n\right)\left(m^{2}-4mn+4n^{2}\right)+2mn\left(m+2n\right)+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Use the distributive property to multiply m+2n by m^{2}-4mn+4n^{2} and combine like terms.

$$m^{3}-2nm^{2}-4mn^{2}+8n^{3}+2mn\left(m+2n\right)+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Use the distributive property to multiply 2mn by m+2n.

$$m^{3}-2nm^{2}-4mn^{2}+8n^{3}+2nm^{2}+4mn^{2}+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Combine -2nm^{2} and 2nm^{2} to get 0.

$$m^{3}-4mn^{2}+8n^{3}+4mn^{2}+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Combine -4mn^{2} and 4mn^{2} to get 0.

$$m^{3}+8n^{3}+\left(m-2n\right)\left(m^{2}+4n^{2}+2mn\right)$$

Use the distributive property to multiply m-2n by m^{2}+4n^{2}+2mn and combine like terms.

$$m^{3}+8n^{3}+m^{3}-8n^{3}$$

Combine m^{3} and m^{3} to get 2m^{3}.

$$2m^{3}+8n^{3}-8n^{3}$$

Combine 8n^{3} and -8n^{3} to get 0.

$$2m^{3}$$