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Factor \(\displaystyle 24 x^{3} + 30 x^{2} - 72 x - 90\).

Factoring out the GCF \(\displaystyle 6\) from each term gives \(\displaystyle 6(4 x^{3} + 5 x^{2} - 12 x - 15)\). Grouping the first two terms and factoring out their GCF, \(\displaystyle x^{2}\), gives \(\displaystyle x^{2}(4 x + 5)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -3\), gives \(\displaystyle -3(4 x + 5)\). The polynomial now has a common binomial factor of \(\displaystyle 4 x + 5\). This gives \(\displaystyle 6[x^{2} \left(4 x + 5\right) -3 \cdot \left(4 x + 5\right)] = 6\left(4 x + 5\right) \left(x^{2} - 3\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $24 x^{3} + 30 x^{2} - 72 x - 90$. 
    \soln{9cm}{Factoring out the GCF $6$ from each term gives $6(4 x^{3} + 5 x^{2} - 12 x - 15)$. Grouping the first two terms and factoring out their GCF, $x^{2}$, gives $x^{2}(4 x + 5)$. Grouping the last two terms and factoring out their GCF, $-3$, gives $-3(4 x + 5)$. The polynomial now has a common binomial factor of $4 x + 5$. This gives $6[x^{2} \left(4 x + 5\right) -3 \cdot \left(4 x + 5\right)] = 6\left(4 x + 5\right) \left(x^{2} - 3\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 24 x^{3} + 30 x^{2} - 72 x - 90 " src="/equation_images/%20%5Cdisplaystyle%2024%20x%5E%7B3%7D%20%2B%2030%20x%5E%7B2%7D%20-%2072%20x%20-%2090%20" alt="LaTeX:  \displaystyle 24 x^{3} + 30 x^{2} - 72 x - 90 " data-equation-content=" \displaystyle 24 x^{3} + 30 x^{2} - 72 x - 90 " /> . </p> </p>
HTML for Canvas
<p> <p>Factoring out the GCF  <img class="equation_image" title=" \displaystyle 6 " src="/equation_images/%20%5Cdisplaystyle%206%20" alt="LaTeX:  \displaystyle 6 " data-equation-content=" \displaystyle 6 " />  from each term gives  <img class="equation_image" title=" \displaystyle 6(4 x^{3} + 5 x^{2} - 12 x - 15) " src="/equation_images/%20%5Cdisplaystyle%206%284%20x%5E%7B3%7D%20%2B%205%20x%5E%7B2%7D%20-%2012%20x%20-%2015%29%20" alt="LaTeX:  \displaystyle 6(4 x^{3} + 5 x^{2} - 12 x - 15) " data-equation-content=" \displaystyle 6(4 x^{3} + 5 x^{2} - 12 x - 15) " /> . 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}(4 x + 5) " src="/equation_images/%20%5Cdisplaystyle%20x%5E%7B2%7D%284%20x%20%2B%205%29%20" alt="LaTeX:  \displaystyle x^{2}(4 x + 5) " data-equation-content=" \displaystyle x^{2}(4 x + 5) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -3 " src="/equation_images/%20%5Cdisplaystyle%20-3%20" alt="LaTeX:  \displaystyle -3 " data-equation-content=" \displaystyle -3 " /> , gives  <img class="equation_image" title=" \displaystyle -3(4 x + 5) " src="/equation_images/%20%5Cdisplaystyle%20-3%284%20x%20%2B%205%29%20" alt="LaTeX:  \displaystyle -3(4 x + 5) " data-equation-content=" \displaystyle -3(4 x + 5) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 4 x + 5 " src="/equation_images/%20%5Cdisplaystyle%204%20x%20%2B%205%20" alt="LaTeX:  \displaystyle 4 x + 5 " data-equation-content=" \displaystyle 4 x + 5 " /> . This gives  <img class="equation_image" title=" \displaystyle 6[x^{2} \left(4 x + 5\right) -3 \cdot \left(4 x + 5\right)] = 6\left(4 x + 5\right) \left(x^{2} - 3\right) " src="/equation_images/%20%5Cdisplaystyle%206%5Bx%5E%7B2%7D%20%5Cleft%284%20x%20%2B%205%5Cright%29%20-3%20%5Ccdot%20%5Cleft%284%20x%20%2B%205%5Cright%29%5D%20%3D%206%5Cleft%284%20x%20%2B%205%5Cright%29%20%5Cleft%28x%5E%7B2%7D%20-%203%5Cright%29%20" alt="LaTeX:  \displaystyle 6[x^{2} \left(4 x + 5\right) -3 \cdot \left(4 x + 5\right)] = 6\left(4 x + 5\right) \left(x^{2} - 3\right) " data-equation-content=" \displaystyle 6[x^{2} \left(4 x + 5\right) -3 \cdot \left(4 x + 5\right)] = 6\left(4 x + 5\right) \left(x^{2} - 3\right) " /> . </p> </p>