Solve (m+1)^3-(m+1)^2+3(m+1)+5

$$( m + 1 ) ^ { 3 } - ( m + 1 ) ^ { 2 } + 3 ( m + 1 ) + 5$$

Evaluate

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

Solution Steps

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

$$m^{3}+3m^{2}+3m+1-\left(m+1\right)^{2}+3\left(m+1\right)+5$$

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

$$m^{3}+3m^{2}+3m+1-\left(m^{2}+2m+1\right)+3\left(m+1\right)+5$$

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

$$m^{3}+3m^{2}+3m+1-m^{2}-2m-1+3\left(m+1\right)+5$$

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

$$m^{3}+2m^{2}+3m+1-2m-1+3\left(m+1\right)+5$$

Combine 3m and -2m to get m.

$$m^{3}+2m^{2}+m+1-1+3\left(m+1\right)+5$$

Subtract 1 from 1 to get 0.

$$m^{3}+2m^{2}+m+3\left(m+1\right)+5$$

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

$$m^{3}+2m^{2}+m+3m+3+5$$

Combine m and 3m to get 4m.

$$m^{3}+2m^{2}+4m+3+5$$

Add 3 and 5 to get 8.

$$m^{3}+2m^{2}+4m+8$$

Expand

$$m^{3}+2m^{2}+4m+8$$

Solution Steps

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

$$m^{3}+3m^{2}+3m+1-\left(m+1\right)^{2}+3\left(m+1\right)+5$$

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

$$m^{3}+3m^{2}+3m+1-\left(m^{2}+2m+1\right)+3\left(m+1\right)+5$$

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

$$m^{3}+3m^{2}+3m+1-m^{2}-2m-1+3\left(m+1\right)+5$$

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

$$m^{3}+2m^{2}+3m+1-2m-1+3\left(m+1\right)+5$$

Combine 3m and -2m to get m.

$$m^{3}+2m^{2}+m+1-1+3\left(m+1\right)+5$$

Subtract 1 from 1 to get 0.

$$m^{3}+2m^{2}+m+3\left(m+1\right)+5$$

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

$$m^{3}+2m^{2}+m+3m+3+5$$

Combine m and 3m to get 4m.

$$m^{3}+2m^{2}+4m+3+5$$

Add 3 and 5 to get 8.

$$m^{3}+2m^{2}+4m+8$$