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Factoring out the GCF \(\displaystyle -9\) from each term gives \(\displaystyle -9(7 x^{3} + 8 x^{2} - 7 x - 8)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(7 x + 8)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -1\), gives \(\displaystyle -1(7 x + 8)\). The polynomial now has a common binomial factor of \(\displaystyle 7 x + 8\). This gives \(\displaystyle -9[x^{2} \left(7 x + 8\right) -1 \cdot \left(7 x + 8\right)] = -9\left(7 x + 8\right) \left(x^{2} - 1\right)\). The quadratic factor can be factored using the difference of squares to give \(\displaystyle -9\left(x - 1\right) \left(x + 1\right) \left(7 x + 8\right). \)

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\begin{question}Factor $- 63 x^{3} - 72 x^{2} + 63 x + 72$. 
    \soln{9cm}{Factoring out the GCF $-9$ from each term gives $-9(7 x^{3} + 8 x^{2} - 7 x - 8)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(7 x + 8)$. Grouping the last two terms and factoring out their GCF, $-1$, gives $-1(7 x + 8)$. The polynomial now has a common binomial factor of $7 x + 8$. This gives $-9[x^{2} \left(7 x + 8\right) -1 \cdot \left(7 x + 8\right)] = -9\left(7 x + 8\right) \left(x^{2} - 1\right)$. The quadratic factor can be factored using the difference of squares to give $-9\left(x - 1\right) \left(x + 1\right) \left(7 x + 8\right). $}

\end{question}

Download Question and Solution Environment\(\LaTeX\)

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\begin{document}\begin{question}(10pts) The question goes here!
    \soln{9cm}{The solution goes here.}

\end{question}\end{document}

HTML for Canvas

<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 63 x^{3} - 72 x^{2} + 63 x + 72 " src="/equation_images/%20%5Cdisplaystyle%20-%2063%20x%5E%7B3%7D%20-%2072%20x%5E%7B2%7D%20%2B%2063%20x%20%2B%2072%20" alt="LaTeX:  \displaystyle - 63 x^{3} - 72 x^{2} + 63 x + 72 " data-equation-content=" \displaystyle - 63 x^{3} - 72 x^{2} + 63 x + 72 " /> . </p> </p>

HTML for Canvas

<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle -9 " src="/equation_images/%20%5Cdisplaystyle%20-9%20" alt="LaTeX:  \displaystyle -9 " data-equation-content=" \displaystyle -9 " />  from each term gives  <img class="equation_image" title=" \displaystyle -9(7 x^{3} + 8 x^{2} - 7 x - 8) " src="/equation_images/%20%5Cdisplaystyle%20-9%287%20x%5E%7B3%7D%20%2B%208%20x%5E%7B2%7D%20-%207%20x%20-%208%29%20" alt="LaTeX:  \displaystyle -9(7 x^{3} + 8 x^{2} - 7 x - 8) " data-equation-content=" \displaystyle -9(7 x^{3} + 8 x^{2} - 7 x - 8) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle x^{2} " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle x^{2} " data-equation-content=" \displaystyle x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle x^{2}(7 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%287%20x%20%2B%208%29%20" alt="LaTeX:  \displaystyle x^{2}(7 x + 8) " data-equation-content=" \displaystyle x^{2}(7 x + 8) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -1 " src="/equation_images/%20%5Cdisplaystyle%20-1%20" alt="LaTeX:  \displaystyle -1 " data-equation-content=" \displaystyle -1 " /> , gives  <img class="equation_image" title=" \displaystyle -1(7 x + 8) " src="/equation_images/%20%5Cdisplaystyle%20-1%287%20x%20%2B%208%29%20" alt="LaTeX:  \displaystyle -1(7 x + 8) " data-equation-content=" \displaystyle -1(7 x + 8) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 7 x + 8 " src="/equation_images/%20%5Cdisplaystyle%207%20x%20%2B%208%20" alt="LaTeX:  \displaystyle 7 x + 8 " data-equation-content=" \displaystyle 7 x + 8 " /> . This gives  <img class="equation_image" title=" \displaystyle -9[x^{2} \left(7 x + 8\right) -1 \cdot \left(7 x + 8\right)] = -9\left(7 x + 8\right) \left(x^{2} - 1\right) " src="/equation_images/%20%5Cdisplaystyle%20-9%5Bx%5E%7B2%7D%20%5Cleft%287%20x%20%2B%208%5Cright%29%20-1%20%5Ccdot%20%5Cleft%287%20x%20%2B%208%5Cright%29%5D%20%3D%20-9%5Cleft%287%20x%20%2B%208%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%201%5Cright%29%20" alt="LaTeX:  \displaystyle -9[x^{2} \left(7 x + 8\right) -1 \cdot \left(7 x + 8\right)] = -9\left(7 x + 8\right) \left(x^{2} - 1\right) " data-equation-content=" \displaystyle -9[x^{2} \left(7 x + 8\right) -1 \cdot \left(7 x + 8\right)] = -9\left(7 x + 8\right) \left(x^{2} - 1\right) " /> . The quadratic factor can be factored using the difference of squares to give  <img class="equation_image" title=" \displaystyle -9\left(x - 1\right) \left(x + 1\right) \left(7 x + 8\right).  " src="/equation_images/%20%5Cdisplaystyle%20-9%5Cleft%28x%20-%201%5Cright%29%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%287%20x%20%2B%208%5Cright%29.%20%20" alt="LaTeX:  \displaystyle -9\left(x - 1\right) \left(x + 1\right) \left(7 x + 8\right).  " data-equation-content=" \displaystyle -9\left(x - 1\right) \left(x + 1\right) \left(7 x + 8\right).  " /> </p> </p>