Solve (m-2n)^3-(2n+m)^3+4n(3m^2+5n^2)

$$( m - 2 n ) ^ { 3 } - ( 2 n + m ) ^ { 3 } + 4 n ( 3 m ^ { 2 } + 5 n ^ { 2 } )$$

Evaluate

$$4n^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(m-2n\right)^{3}.

$$m^{3}-6m^{2}n+12mn^{2}-8n^{3}-\left(2n+m\right)^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2n+m\right)^{3}.

$$m^{3}-6m^{2}n+12mn^{2}-8n^{3}-\left(8n^{3}+12n^{2}m+6nm^{2}+m^{3}\right)+4n\left(3m^{2}+5n^{2}\right)$$

To find the opposite of 8n^{3}+12n^{2}m+6nm^{2}+m^{3}, find the opposite of each term.

$$m^{3}-6m^{2}n+12mn^{2}-8n^{3}-8n^{3}-12n^{2}m-6nm^{2}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine -8n^{3} and -8n^{3} to get -16n^{3}.

$$m^{3}-6m^{2}n+12mn^{2}-16n^{3}-12n^{2}m-6nm^{2}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine 12mn^{2} and -12n^{2}m to get 0.

$$m^{3}-6m^{2}n-16n^{3}-6nm^{2}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine -6m^{2}n and -6nm^{2} to get -12m^{2}n.

$$m^{3}-12m^{2}n-16n^{3}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine m^{3} and -m^{3} to get 0.

$$-12m^{2}n-16n^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Use the distributive property to multiply 4n by 3m^{2}+5n^{2}.

$$-12m^{2}n-16n^{3}+12nm^{2}+20n^{3}$$

Combine -12m^{2}n and 12nm^{2} to get 0.

$$-16n^{3}+20n^{3}$$

Combine -16n^{3} and 20n^{3} to get 4n^{3}.

$$4n^{3}$$

Expand

$$4n^{3}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(m-2n\right)^{3}.

$$m^{3}-6m^{2}n+12mn^{2}-8n^{3}-\left(2n+m\right)^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(2n+m\right)^{3}.

$$m^{3}-6m^{2}n+12mn^{2}-8n^{3}-\left(8n^{3}+12n^{2}m+6nm^{2}+m^{3}\right)+4n\left(3m^{2}+5n^{2}\right)$$

To find the opposite of 8n^{3}+12n^{2}m+6nm^{2}+m^{3}, find the opposite of each term.

$$m^{3}-6m^{2}n+12mn^{2}-8n^{3}-8n^{3}-12n^{2}m-6nm^{2}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine -8n^{3} and -8n^{3} to get -16n^{3}.

$$m^{3}-6m^{2}n+12mn^{2}-16n^{3}-12n^{2}m-6nm^{2}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine 12mn^{2} and -12n^{2}m to get 0.

$$m^{3}-6m^{2}n-16n^{3}-6nm^{2}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine -6m^{2}n and -6nm^{2} to get -12m^{2}n.

$$m^{3}-12m^{2}n-16n^{3}-m^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Combine m^{3} and -m^{3} to get 0.

$$-12m^{2}n-16n^{3}+4n\left(3m^{2}+5n^{2}\right)$$

Use the distributive property to multiply 4n by 3m^{2}+5n^{2}.

$$-12m^{2}n-16n^{3}+12nm^{2}+20n^{3}$$

Combine -12m^{2}n and 12nm^{2} to get 0.

$$-16n^{3}+20n^{3}$$

Combine -16n^{3} and 20n^{3} to get 4n^{3}.

$$4n^{3}$$