Solve (-1x-2)^2x^3sqrt{-8}

$$( - 1 x - 2 ) ^ { 2 } x ^ { 3 } \sqrt { - 8 }$$

Evaluate (complex solution)

$$2\sqrt{2}i\left(x+2\right)^{2}x^{3}$$

Solution Steps

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

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

Factor -8=\left(2i\right)^{2}\times 2. Rewrite the square root of the product \sqrt{\left(2i\right)^{2}\times 2} as the product of square roots \sqrt{\left(2i\right)^{2}}\sqrt{2}. Take the square root of \left(2i\right)^{2}.

$$\left(x^{2}+4x+4\right)x^{3}\times \left(2i\right)\sqrt{2}$$

Use the distributive property to multiply x^{2}+4x+4 by x^{3}.

$$\left(x^{5}+4x^{4}+4x^{3}\right)\times \left(2i\right)\sqrt{2}$$

Use the distributive property to multiply x^{5}+4x^{4}+4x^{3} by 2i.

$$\left(2ix^{5}+8ix^{4}+8ix^{3}\right)\sqrt{2}$$

Use the distributive property to multiply 2ix^{5}+8ix^{4}+8ix^{3} by \sqrt{2}.

$$2ix^{5}\sqrt{2}+8ix^{4}\sqrt{2}+8ix^{3}\sqrt{2}$$

Evaluate

$$\text{Indeterminate}$$

Solution Steps

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

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

Use the distributive property to multiply x^{2}+4x+4 by x^{3}.

$$\left(x^{5}+4x^{4}+4x^{3}\right)\sqrt{-8}$$

Use the distributive property to multiply x^{5}+4x^{4}+4x^{3} by \sqrt{-8}.

$$x^{5}\sqrt{-8}+4x^{4}\sqrt{-8}+4x^{3}\sqrt{-8}$$