Answer
$$(x+1)(x+2)^2(x+3)=x^4+8x^3+23x^2+28x+12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)(x+2)^2(x+3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x+1)(x^2+4x+4)(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3+4x^2+4x+x^2+4x+4)(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3+5x^2+8x+4)(x+3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4+8x^3+23x^2+28x+12\end{aligned} $$ | |
| 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 each term of $ \left( \color{blue}{x+1}\right) $ by each term in $ \left( x^2+4x+4\right) $. $$ \left( \color{blue}{x+1}\right) \cdot \left( x^2+4x+4\right) = x^3+4x^2+4x+x^2+4x+4 $$ |
| ③ | Combine like terms: $$ x^3+ \color{blue}{4x^2} + \color{red}{4x} + \color{blue}{x^2} + \color{red}{4x} +4 = x^3+ \color{blue}{5x^2} + \color{red}{8x} +4 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3+5x^2+8x+4}\right) $ by each term in $ \left( x+3\right) $. $$ \left( \color{blue}{x^3+5x^2+8x+4}\right) \cdot \left( x+3\right) = x^4+3x^3+5x^3+15x^2+8x^2+24x+4x+12 $$ |
| ⑤ | Combine like terms: $$ x^4+ \color{blue}{3x^3} + \color{blue}{5x^3} + \color{red}{15x^2} + \color{red}{8x^2} + \color{green}{24x} + \color{green}{4x} +12 = x^4+ \color{blue}{8x^3} + \color{red}{23x^2} + \color{green}{28x} +12 $$ |
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