Answer
$$(x^2+3x+2)(x^2-2x-1)(x^2-7x+12)=x^6-6x^5+40x^3-13x^2-70x-24$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^2+3x+2)(x^2-2x-1)(x^2-7x+12)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^4+x^3-5x^2-7x-2)(x^2-7x+12) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^6-6x^5+40x^3-13x^2-70x-24\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^2+3x+2}\right) $ by each term in $ \left( x^2-2x-1\right) $. $$ \left( \color{blue}{x^2+3x+2}\right) \cdot \left( x^2-2x-1\right) = x^4-2x^3-x^2+3x^3-6x^2-3x+2x^2-4x-2 $$ |
| ② | Combine like terms: $$ x^4 \color{blue}{-2x^3} \color{red}{-x^2} + \color{blue}{3x^3} \color{green}{-6x^2} \color{orange}{-3x} + \color{green}{2x^2} \color{orange}{-4x} -2 = x^4+ \color{blue}{x^3} \color{green}{-5x^2} \color{orange}{-7x} -2 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^4+x^3-5x^2-7x-2}\right) $ by each term in $ \left( x^2-7x+12\right) $. $$ \left( \color{blue}{x^4+x^3-5x^2-7x-2}\right) \cdot \left( x^2-7x+12\right) = \\ = x^6-7x^5+12x^4+x^5-7x^4+12x^3-5x^4+35x^3-60x^2-7x^3+49x^2-84x-2x^2+14x-24 $$ |
| ④ | Combine like terms: $$ x^6 \color{blue}{-7x^5} + \color{red}{12x^4} + \color{blue}{x^5} \color{green}{-7x^4} + \color{orange}{12x^3} \color{green}{-5x^4} + \color{blue}{35x^3} \color{red}{-60x^2} \color{blue}{-7x^3} + \color{green}{49x^2} \color{orange}{-84x} \color{green}{-2x^2} + \color{orange}{14x} -24 = \\ = x^6 \color{blue}{-6x^5} + \color{blue}{40x^3} \color{green}{-13x^2} \color{orange}{-70x} -24 $$ |
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