Solve 6x^{2}(x+1)^-frac{1{3}}(x-2)^3-3x(x+1)^frac{2{3}}(x-2)^2

$$6 x ^ { 2 } ( x + 1 ) ^ { - \frac { 1 } { 3 } } ( x - 2 ) ^ { 3 } - 3 x ( x + 1 ) ^ { \frac { 2 } { 3 } } ( x - 2 ) ^ { 2 }$$

Factor

$$\frac{6x\left(x-\frac{5-\sqrt{33}}{4}\right)\left(x-\frac{\sqrt{33}+5}{4}\right)\left(x-2\right)^{2}}{\sqrt[3]{x+1}}$$

Solution Steps

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

$$factor(6x^{2}\left(x+1\right)^{-\frac{1}{3}}\left(x^{3}-6x^{2}+12x-8\right)-3x\left(x+1\right)^{\frac{2}{3}}\left(x-2\right)^{2})$$

Use the distributive property to multiply 6x^{2}\left(x+1\right)^{-\frac{1}{3}} by x^{3}-6x^{2}+12x-8.

$$factor(6\left(x+1\right)^{-\frac{1}{3}}x^{5}-36\left(x+1\right)^{-\frac{1}{3}}x^{4}+72\left(x+1\right)^{-\frac{1}{3}}x^{3}-48\left(x+1\right)^{-\frac{1}{3}}x^{2}-3x\left(x+1\right)^{\frac{2}{3}}\left(x-2\right)^{2})$$

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

$$factor(6\left(x+1\right)^{-\frac{1}{3}}x^{5}-36\left(x+1\right)^{-\frac{1}{3}}x^{4}+72\left(x+1\right)^{-\frac{1}{3}}x^{3}-48\left(x+1\right)^{-\frac{1}{3}}x^{2}-3x\left(x+1\right)^{\frac{2}{3}}\left(x^{2}-4x+4\right))$$

Use the distributive property to multiply -3x\left(x+1\right)^{\frac{2}{3}} by x^{2}-4x+4.

$$factor(6\left(x+1\right)^{-\frac{1}{3}}x^{5}-36\left(x+1\right)^{-\frac{1}{3}}x^{4}+72\left(x+1\right)^{-\frac{1}{3}}x^{3}-48\left(x+1\right)^{-\frac{1}{3}}x^{2}-3\left(x+1\right)^{\frac{2}{3}}x^{3}+12\left(x+1\right)^{\frac{2}{3}}x^{2}-12\left(x+1\right)^{\frac{2}{3}}x)$$

Factor out 3.

$$3\left(\frac{2x^{5}}{\sqrt[3]{x+1}}-\frac{12x^{4}}{\sqrt[3]{x+1}}+\frac{24x^{3}}{\sqrt[3]{x+1}}-\frac{16x^{2}}{\sqrt[3]{x+1}}-\left(x+1\right)^{\frac{2}{3}}x^{3}+4\left(x+1\right)^{\frac{2}{3}}x^{2}-4\left(x+1\right)^{\frac{2}{3}}x\right)$$

Consider 2\left(x+1\right)^{-\frac{1}{3}}x^{5}-12\left(x+1\right)^{-\frac{1}{3}}x^{4}+24\left(x+1\right)^{-\frac{1}{3}}x^{3}-16\left(x+1\right)^{-\frac{1}{3}}x^{2}-\left(x+1\right)^{\frac{2}{3}}x^{3}+4\left(x+1\right)^{\frac{2}{3}}x^{2}-4\left(x+1\right)^{\frac{2}{3}}x. Factor out x.

$$x\left(\frac{2x^{4}}{\sqrt[3]{x+1}}-\frac{12x^{3}}{\sqrt[3]{x+1}}+\frac{24x^{2}}{\sqrt[3]{x+1}}-\frac{16x}{\sqrt[3]{x+1}}-\left(x+1\right)^{\frac{2}{3}}x^{2}+4\left(x+1\right)^{\frac{2}{3}}x-4\left(x+1\right)^{\frac{2}{3}}\right)$$

Rewrite the complete factored expression.

$$3x\left(\frac{2x^{4}}{\sqrt[3]{x+1}}-\frac{12x^{3}}{\sqrt[3]{x+1}}+\frac{24x^{2}}{\sqrt[3]{x+1}}-\frac{16x}{\sqrt[3]{x+1}}-\left(x+1\right)^{\frac{2}{3}}x^{2}+4\left(x+1\right)^{\frac{2}{3}}x-4\left(x+1\right)^{\frac{2}{3}}\right)$$

Evaluate

$$\frac{3x\left(x-2\right)^{2}\left(2x^{2}-5x-1\right)}{\sqrt[3]{x+1}}$$