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Factor \(\displaystyle 42 x^{3} + 54 x^{2} - 35 x - 45\).

Grouping the first two terms and factoring out their GCF, \(\displaystyle 6 x^{2}\), gives \(\displaystyle 6 x^{2}(7 x + 9)\). Grouping the last two terms and factoring out their GCF, \(\displaystyle -5\), gives \(\displaystyle -5(7 x + 9)\). The polynomial now has a common binomial factor of \(\displaystyle 7 x + 9\). This gives \(\displaystyle 6 x^{2} \left(7 x + 9\right) -5 \cdot \left(7 x + 9\right) = \left(7 x + 9\right) \left(6 x^{2} - 5\right)\).

Download \(\LaTeX\) 

\begin{question}Factor $42 x^{3} + 54 x^{2} - 35 x - 45$. 
    \soln{9cm}{Grouping the first two terms and factoring out their GCF, $6 x^{2}$, gives $6 x^{2}(7 x + 9)$. Grouping the last two terms and factoring out their GCF, $-5$, gives $-5(7 x + 9)$. The polynomial now has a common binomial factor of $7 x + 9$. This gives $6 x^{2} \left(7 x + 9\right) -5 \cdot \left(7 x + 9\right) = \left(7 x + 9\right) \left(6 x^{2} - 5\right)$. }

\end{question}

Download Question and Solution Environment\(\LaTeX\)
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HTML for Canvas 
<p> <p>Factor  <img class="equation_image" title=" \displaystyle 42 x^{3} + 54 x^{2} - 35 x - 45 " src="/equation_images/%20%5Cdisplaystyle%2042%20x%5E%7B3%7D%20%2B%2054%20x%5E%7B2%7D%20-%2035%20x%20-%2045%20" alt="LaTeX:  \displaystyle 42 x^{3} + 54 x^{2} - 35 x - 45 " data-equation-content=" \displaystyle 42 x^{3} + 54 x^{2} - 35 x - 45 " /> . </p> </p>
HTML for Canvas
<p> <p>Grouping the first two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle 6 x^{2} " src="/equation_images/%20%5Cdisplaystyle%206%20x%5E%7B2%7D%20" alt="LaTeX:  \displaystyle 6 x^{2} " data-equation-content=" \displaystyle 6 x^{2} " /> , gives  <img class="equation_image" title=" \displaystyle 6 x^{2}(7 x + 9) " src="/equation_images/%20%5Cdisplaystyle%206%20x%5E%7B2%7D%287%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle 6 x^{2}(7 x + 9) " data-equation-content=" \displaystyle 6 x^{2}(7 x + 9) " /> . Grouping the last two terms and factoring out their GCF,  <img class="equation_image" title=" \displaystyle -5 " src="/equation_images/%20%5Cdisplaystyle%20-5%20" alt="LaTeX:  \displaystyle -5 " data-equation-content=" \displaystyle -5 " /> , gives  <img class="equation_image" title=" \displaystyle -5(7 x + 9) " src="/equation_images/%20%5Cdisplaystyle%20-5%287%20x%20%2B%209%29%20" alt="LaTeX:  \displaystyle -5(7 x + 9) " data-equation-content=" \displaystyle -5(7 x + 9) " /> . The polynomial now has a common binomial factor of  <img class="equation_image" title=" \displaystyle 7 x + 9 " src="/equation_images/%20%5Cdisplaystyle%207%20x%20%2B%209%20" alt="LaTeX:  \displaystyle 7 x + 9 " data-equation-content=" \displaystyle 7 x + 9 " /> . This gives  <img class="equation_image" title=" \displaystyle 6 x^{2} \left(7 x + 9\right) -5 \cdot \left(7 x + 9\right) = \left(7 x + 9\right) \left(6 x^{2} - 5\right) " src="/equation_images/%20%5Cdisplaystyle%206%20x%5E%7B2%7D%20%5Cleft%287%20x%20%2B%209%5Cright%29%20-5%20%5Ccdot%20%5Cleft%287%20x%20%2B%209%5Cright%29%20%3D%20%5Cleft%287%20x%20%2B%209%5Cright%29%20%5Cleft%286%20x%5E%7B2%7D%20-%205%5Cright%29%20" alt="LaTeX:  \displaystyle 6 x^{2} \left(7 x + 9\right) -5 \cdot \left(7 x + 9\right) = \left(7 x + 9\right) \left(6 x^{2} - 5\right) " data-equation-content=" \displaystyle 6 x^{2} \left(7 x + 9\right) -5 \cdot \left(7 x + 9\right) = \left(7 x + 9\right) \left(6 x^{2} - 5\right) " /> . </p> </p>