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Question

# How do you factor $3{x}^{2}+8x+4$ by using the grouping method?

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Solution

## Factor the quadratic polynomial $\mathbf{3}{\mathbit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{8}\mathbit{x}\mathbf{+}\mathbf{4}$ by using the grouping method:Find the product of the quadratic term $3{x}^{2}$ and the constant term $4$, $3{x}^{2}×4=12{x}^{2}$ . By a hit and trial process, we have to find two factors that give this product $12{x}^{2}$, such that the factors add up to the linear term $8x$. The two factors that satisfy these conditions are $2x$ and $6x$, because their sum gives the linear term $8x$ and their product gives $12{x}^{2}$.Rewrite $8x$ as $6x+2x$ in the expression $3{x}^{2}+8x+4$. Then factorize by grouping the terms.$\begin{array}{rcl}3{x}^{2}+8x+4& =& 3{x}^{2}+6x+2x+4\\ & =& \left(3{x}^{2}+6x\right)+\left(2x+4\right)\\ & =& 3x\left(x+2\right)+2\left(x+2\right)\\ & =& \left(x+2\right)\left(3x+2\right)\end{array}$Therefore, $3{x}^{2}+8x+4$ is factorized to $\mathbf{\left(}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{\right)}\mathbf{\left(}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{\right)}$ by using the grouping method as shown above.

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