Solve (x+1)(x+2)^2(x^2+1)

$$( x + 1 ) ( x + 2 ) ^ { 2 } ( x ^ { 2 } + 1 )$$

Expand

$$x^{5}+5x^{4}+9x^{3}+9x^{2}+8x+4$$

Solution Steps

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

$$\left(x+1\right)\left(x^{2}+4x+4\right)\left(x^{2}+1\right)$$

Use the distributive property to multiply x+1 by x^{2}+4x+4 and combine like terms.

$$\left(x^{3}+5x^{2}+8x+4\right)\left(x^{2}+1\right)$$

Use the distributive property to multiply x^{3}+5x^{2}+8x+4 by x^{2}+1 and combine like terms.

$$x^{5}+9x^{3}+5x^{4}+9x^{2}+8x+4$$

Evaluate

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

Solution Steps

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

$$\left(x+1\right)\left(x^{2}+4x+4\right)\left(x^{2}+1\right)$$

Use the distributive property to multiply x+1 by x^{2}+4x+4 and combine like terms.

$$\left(x^{3}+5x^{2}+8x+4\right)\left(x^{2}+1\right)$$

Use the distributive property to multiply x^{3}+5x^{2}+8x+4 by x^{2}+1 and combine like terms.

$$x^{5}+9x^{3}+5x^{4}+9x^{2}+8x+4$$