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Factor \(\displaystyle - 90 x^{3} - 60 x^{2} - 45 x - 30\).

Factoring out the GCF \(\displaystyle -15\) from each term gives \(\displaystyle -15(6 x^{3} + 4 x^{2} + 3 x + 2)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle 2 x^{2}\), gives \(\displaystyle 2 x^{2}(3 x + 2)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle 1\), gives \(\displaystyle 1(3 x + 2)\). The polynomial now has a common binomial factor of \(\displaystyle 3 x + 2\). This gives \(\displaystyle -15[2 x^{2} \left(3 x + 2\right) +1 \cdot \left(3 x + 2\right)] = -15\left(3 x + 2\right) \left(2 x^{2} + 1\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $- 90 x^{3} - 60 x^{2} - 45 x - 30$. 
    \soln{9cm}{Factoring out the GCF $-15$ from each term gives $-15(6 x^{3} + 4 x^{2} + 3 x + 2)$. Grouping the first two terms and factoring out their GCF, $2 x^{2}$, gives $2 x^{2}(3 x + 2)$. Grouping the last two terms and factoring out their GCF, $1$, gives $1(3 x + 2)$. The polynomial now has a common binomial factor of $3 x + 2$. This gives $-15[2 x^{2} \left(3 x + 2\right) +1 \cdot \left(3 x + 2\right)] = -15\left(3 x + 2\right) \left(2 x^{2} + 1\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle - 90 x^{3} - 60 x^{2} - 45 x - 30 " src="/equation_images/%20%5Cdisplaystyle%20-%2090%20x%5E%7B3%7D%20-%2060%20x%5E%7B2%7D%20-%2045%20x%20-%2030%20" alt="LaTeX:  \displaystyle - 90 x^{3} - 60 x^{2} - 45 x - 30 " data-equation-content=" \displaystyle - 90 x^{3} - 60 x^{2} - 45 x - 30 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle -15 " src="/equation_images/%20%5Cdisplaystyle%20-15%20" alt="LaTeX:  \displaystyle -15 " data-equation-content=" \displaystyle -15 " />  from each term gives  <img class="equation_image" title=" \displaystyle -15(6 x^{3} + 4 x^{2} + 3 x + 2) " src="/equation_images/%20%5Cdisplaystyle%20-15%286%20x%5E%7B3%7D%20%2B%204%20x%5E%7B2%7D%20%2B%203%20x%20%2B%202%29%20" alt="LaTeX:  \displaystyle -15(6 x^{3} + 4 x^{2} + 3 x + 2) " data-equation-content=" \displaystyle -15(6 x^{3} + 4 x^{2} + 3 x + 2) " /> . Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 2 x^{2} " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 2 x^{2} " data-equation-content=" \displaystyle 2 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 2 x^{2}(3 x + 2) " src="/equation_images/%20%5Cdisplaystyle%202%20x%5E%7B2%7D%283%20x%20%2B%202%29%20" alt="LaTeX:  \displaystyle 2 x^{2}(3 x + 2) " data-equation-content=" \displaystyle 2 x^{2}(3 x + 2) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 1 " src="/equation_images/%20%5Cdisplaystyle%201%20" alt="LaTeX:  \displaystyle 1 " data-equation-content=" \displaystyle 1 " /> , gives  <img class="equation_image" title=" \displaystyle 1(3 x + 2) " src="/equation_images/%20%5Cdisplaystyle%201%283%20x%20%2B%202%29%20" alt="LaTeX:  \displaystyle 1(3 x + 2) " data-equation-content=" \displaystyle 1(3 x + 2) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 3 x + 2 " src="/equation_images/%20%5Cdisplaystyle%203%20x%20%2B%202%20" alt="LaTeX:  \displaystyle 3 x + 2 " data-equation-content=" \displaystyle 3 x + 2 " /> . This gives  <img class="equation_image" title=" \displaystyle -15[2 x^{2} \left(3 x + 2\right) +1 \cdot \left(3 x + 2\right)] = -15\left(3 x + 2\right) \left(2 x^{2} + 1\right) " src="/equation_images/%20%5Cdisplaystyle%20-15%5B2%20x%5E%7B2%7D%20%5Cleft%283%20x%20%2B%202%5Cright%29%20%2B1%20%5Ccdot%20%5Cleft%283%20x%20%2B%202%5Cright%29%5D%20%3D%20-15%5Cleft%283%20x%20%2B%202%5Cright%29%20%5Cleft%282%20x%5E%7B2%7D%20%2B%201%5Cright%29%20" alt="LaTeX:  \displaystyle -15[2 x^{2} \left(3 x + 2\right) +1 \cdot \left(3 x + 2\right)] = -15\left(3 x + 2\right) \left(2 x^{2} + 1\right) " data-equation-content=" \displaystyle -15[2 x^{2} \left(3 x + 2\right) +1 \cdot \left(3 x + 2\right)] = -15\left(3 x + 2\right) \left(2 x^{2} + 1\right) " /> . </p> </p>