Answer
$$x(x+1)(x-3)(x-5)=x^4-7x^3+7x^2+15x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x+1)(x-3)(x-5)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+x)(x-3)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-3x^2+x^2-3x)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-2x^2-3x)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^4-5x^3-2x^3+10x^2-3x^2+15x \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-7x^3+7x^2+15x\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( x+1\right) $ $$ \color{blue}{x} \cdot \left( x+1\right) = x^2+x $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+x}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^2+x}\right) \cdot \left( x-3\right) = x^3-3x^2+x^2-3x $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-3x^2} + \color{blue}{x^2} -3x = x^3 \color{blue}{-2x^2} -3x $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-2x^2-3x}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{x^3-2x^2-3x}\right) \cdot \left( x-5\right) = x^4-5x^3-2x^3+10x^2-3x^2+15x $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-5x^3} \color{blue}{-2x^3} + \color{red}{10x^2} \color{red}{-3x^2} +15x = x^4 \color{blue}{-7x^3} + \color{red}{7x^2} +15x $$ |
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