Solve K(K+1)^2+(K+1)^3

$$K ( K + 1 ) ^ { 2 } + ( K + 1 ) ^ { 3 }$$

Evaluate

$$\left(2K+1\right)\left(K+1\right)^{2}$$

Solution Steps

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

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

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

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

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

$$K^{3}+2K^{2}+K+K^{3}+3K^{2}+3K+1$$

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

$$2K^{3}+2K^{2}+K+3K^{2}+3K+1$$

Combine 2K^{2} and 3K^{2} to get 5K^{2}.

$$2K^{3}+5K^{2}+K+3K+1$$

Combine K and 3K to get 4K.

$$2K^{3}+5K^{2}+4K+1$$

Expand

$$2K^{3}+5K^{2}+4K+1$$

Solution Steps

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

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

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

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

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

$$K^{3}+2K^{2}+K+K^{3}+3K^{2}+3K+1$$

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

$$2K^{3}+2K^{2}+K+3K^{2}+3K+1$$

Combine 2K^{2} and 3K^{2} to get 5K^{2}.

$$2K^{3}+5K^{2}+K+3K+1$$

Combine K and 3K to get 4K.

$$2K^{3}+5K^{2}+4K+1$$