Answer
$$(x-1)^2(x+2)^3=x^5+4x^4+x^3-10x^2-4x+8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-1)^2(x+2)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-2x+1)(x^3+6x^2+12x+8) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^5+4x^4+x^3-10x^2-4x+8\end{aligned} $$ | |
| ① | Find $ \left(x-1\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}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$Find $ \left(x+2\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 2 $. $$ \left(x+2\right)^3 = x^3+3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2+2^3 = x^3+6x^2+12x+8 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-2x+1}\right) $ by each term in $ \left( x^3+6x^2+12x+8\right) $. $$ \left( \color{blue}{x^2-2x+1}\right) \cdot \left( x^3+6x^2+12x+8\right) = \\ = x^5+6x^4+ \cancel{12x^3}+8x^2-2x^4 -\cancel{12x^3}-24x^2-16x+x^3+6x^2+12x+8 $$ |
| ③ | Combine like terms: $$ x^5+ \color{blue}{6x^4} + \, \color{red}{ \cancel{12x^3}} \,+ \color{orange}{8x^2} \color{blue}{-2x^4} \, \color{blue}{ -\cancel{12x^3}} \, \color{red}{-24x^2} \color{green}{-16x} + \color{blue}{x^3} + \color{red}{6x^2} + \color{green}{12x} +8 = \\ = x^5+ \color{blue}{4x^4} + \color{blue}{x^3} \color{red}{-10x^2} \color{green}{-4x} +8 $$ |
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