Solve (x^2+4x)^2+8(x^2+4x)+16

$$( x ^ { 2 } + 4 x ) ^ { 2 } + 8 ( x ^ { 2 } + 4 x ) + 16$$

Evaluate

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

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+4x\right)^{2}.

$$\left(x^{2}\right)^{2}+8x^{2}x+16x^{2}+8\left(x^{2}+4x\right)+16$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}+8x^{2}x+16x^{2}+8\left(x^{2}+4x\right)+16$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{4}+8x^{3}+16x^{2}+8\left(x^{2}+4x\right)+16$$

Use the distributive property to multiply 8 by x^{2}+4x.

$$x^{4}+8x^{3}+16x^{2}+8x^{2}+32x+16$$

Combine 16x^{2} and 8x^{2} to get 24x^{2}.

$$x^{4}+8x^{3}+24x^{2}+32x+16$$

Expand

$$x^{4}+8x^{3}+24x^{2}+32x+16$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+4x\right)^{2}.

$$\left(x^{2}\right)^{2}+8x^{2}x+16x^{2}+8\left(x^{2}+4x\right)+16$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}+8x^{2}x+16x^{2}+8\left(x^{2}+4x\right)+16$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$x^{4}+8x^{3}+16x^{2}+8\left(x^{2}+4x\right)+16$$

Use the distributive property to multiply 8 by x^{2}+4x.

$$x^{4}+8x^{3}+16x^{2}+8x^{2}+32x+16$$

Combine 16x^{2} and 8x^{2} to get 24x^{2}.

$$x^{4}+8x^{3}+24x^{2}+32x+16$$