Problem:
Multiply and expand $p(x) = \sqrt{5} x-\sqrt{5} x^2$ and $q(x) = -3 \sqrt{5} x-4 \sqrt{5}$
Answer:
\begin{array}{l}
 
\begin{array}{l}
 \text{Expand the following}: \\
 \left(\sqrt{5} x-\sqrt{5} x^2\right) \left(-3 \sqrt{5} x-4 \sqrt{5}\right) \\
\end{array}
 \\
\hline
 
\begin{array}{l}
 \left(\sqrt{5} x-\sqrt{5} x^2\right) \left(-3 \sqrt{5} x-4 \sqrt{5}\right)=\left(\sqrt{5} x\right) \left(-3 \sqrt{5} x\right)\text{$\, $+$\, $}\left(\sqrt{5} x\right) \left(-4 \sqrt{5}\right)\text{$\, $+$\, $}\left(-\sqrt{5} x^2\right) \left(-3 \sqrt{5} x\right)\text{$\, $+$\, $}\left(-\sqrt{5} x^2\right) \left(-4 \sqrt{5}\right): \\
 -(-3) \sqrt{5} \sqrt{5} x^2 x-4 (-1) \sqrt{5} \sqrt{5} x^2-3 \sqrt{5} \sqrt{5} x x-4 \sqrt{5} \sqrt{5} x \\
\end{array}
 \\
 
\begin{array}{l}
 \sqrt{5} x (-3) \sqrt{5} x=5 x^2 (-3): \\
 -(-3) \sqrt{5} \sqrt{5} x^2 x-4 (-1) \sqrt{5} \sqrt{5} x^2+\fbox{$-15 x^2$}-4 \sqrt{5} \sqrt{5} x \\
\end{array}
 \\
 
\begin{array}{l}
 5 (-3) \text{= }-15: \\
 -(-3) \sqrt{5} \sqrt{5} x^2 x-4 (-1) \sqrt{5} \sqrt{5} x^2+\fbox{$-15$} x^2-4 \sqrt{5} \sqrt{5} x \\
\end{array}
 \\
 
\begin{array}{l}
 -4 \sqrt{5} \sqrt{5} x=-4\times 5 x: \\
 -(-3) \sqrt{5} \sqrt{5} x^2 x-4 (-1) \sqrt{5} \sqrt{5} x^2-15 x^2+\fbox{$-20 x$} \\
\end{array}
 \\
 
\begin{array}{l}
 -4\times 5 \text{= }-20: \\
 -(-3) \sqrt{5} \sqrt{5} x^2 x-4 (-1) \sqrt{5} \sqrt{5} x^2-15 x^2+\fbox{$-20$} x \\
\end{array}
 \\
 
\begin{array}{l}
 -\sqrt{5} x^2 (-3) \sqrt{5} x=-5 x^{2+1} (-3): \\
 \fbox{$15 x^3$}-4 (-1) \sqrt{5} \sqrt{5} x^2-15 x^2-20 x \\
\end{array}
 \\
 
\begin{array}{l}
 2+1=3: \\
 -(-3) 5 x^{\fbox{$3$}}-4 (-1) \sqrt{5} \sqrt{5} x^2-15 x^2-20 x \\
\end{array}
 \\
 
\begin{array}{l}
 -5 (-3) \text{= }15: \\
 \fbox{$15$} x^3-4 (-1) \sqrt{5} \sqrt{5} x^2-15 x^2-20 x \\
\end{array}
 \\
 
\begin{array}{l}
 -4 \sqrt{5} \left(-\sqrt{5}\right) x^2=-4\times 5 \left(-x^2\right): \\
 15 x^3+\fbox{$20 x^2$}-15 x^2-20 x \\
\end{array}
 \\
 
\begin{array}{l}
 -4 (-5) \text{= }20: \\
 15 x^3+\fbox{$20$} x^2-15 x^2-20 x \\
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 \\
 
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 \text{Grouping }\text{like }\text{terms, }15 x^3+20 x^2-15 x^2-20 x=15 x^3+\left(-15 x^2+20 x^2\right)-20 x: \\
 15 x^3+\left(-15 x^2+20 x^2\right)-20 x \\
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 \\
 
\begin{array}{l}
 20 x^2-15 x^2=5 x^2: \\
 \fbox{$
\begin{array}{ll}
 \text{Answer:} &  \\
 \text{} & 15 x^3+\fbox{$5 x^2$}-20 x \\
\end{array}
$} \\
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 \\
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{'source': 'AMPS', 'file': 'amps/mathematica/algebra/multiply_polynomials_w_steps/793.txt'}