Answer
$$x(x-1)(x-2)^2=x^4-5x^3+8x^2-4x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x-1)(x-2)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x(x-1)(x^2-4x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-x)(x^2-4x+4) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^4-4x^3+4x^2-x^3+4x^2-4x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^4-5x^3+8x^2-4x\end{aligned} $$ | |
| ① | Find $ \left(x-2\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 2 }$. $$ \begin{aligned}\left(x-2\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 2 + \color{red}{2^2} = x^2-4x+4\end{aligned} $$ |
| ② | Multiply $ \color{blue}{x} $ by $ \left( x-1\right) $ $$ \color{blue}{x} \cdot \left( x-1\right) = x^2-x $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-x}\right) $ by each term in $ \left( x^2-4x+4\right) $. $$ \left( \color{blue}{x^2-x}\right) \cdot \left( x^2-4x+4\right) = x^4-4x^3+4x^2-x^3+4x^2-4x $$ |
| ④ | Combine like terms: $$ x^4 \color{blue}{-4x^3} + \color{red}{4x^2} \color{blue}{-x^3} + \color{red}{4x^2} -4x = x^4 \color{blue}{-5x^3} + \color{red}{8x^2} -4x $$ |
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