Solve (m-2)^3-(m+1)^3-9(m-m^2-1)

$$( m - 2 ) ^ { 3 } - ( m + 1 ) ^ { 3 } - 9 ( m - m ^ { 2 } - 1 )$$

Evaluate

$$0$$

Solution Steps

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

$$m^{3}-6m^{2}+12m-8-\left(m+1\right)^{3}-9\left(m-m^{2}-1\right)$$

Use binomial theorem \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} to expand \left(m+1\right)^{3}.

$$m^{3}-6m^{2}+12m-8-\left(m^{3}+3m^{2}+3m+1\right)-9\left(m-m^{2}-1\right)$$

To find the opposite of m^{3}+3m^{2}+3m+1, find the opposite of each term.

$$m^{3}-6m^{2}+12m-8-m^{3}-3m^{2}-3m-1-9\left(m-m^{2}-1\right)$$

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

$$-6m^{2}+12m-8-3m^{2}-3m-1-9\left(m-m^{2}-1\right)$$

Combine -6m^{2} and -3m^{2} to get -9m^{2}.

$$-9m^{2}+12m-8-3m-1-9\left(m-m^{2}-1\right)$$

Combine 12m and -3m to get 9m.

$$-9m^{2}+9m-8-1-9\left(m-m^{2}-1\right)$$

Subtract 1 from -8 to get -9.

$$-9m^{2}+9m-9-9\left(m-m^{2}-1\right)$$

Use the distributive property to multiply -9 by m-m^{2}-1.

$$-9m^{2}+9m-9-9m+9m^{2}+9$$

Combine 9m and -9m to get 0.

$$-9m^{2}-9+9m^{2}+9$$

Combine -9m^{2} and 9m^{2} to get 0.

$$-9+9$$

Add -9 and 9 to get 0.

$$0$$

Factor

$$0$$