Question

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

Answer 1

Factor the quadratic polynomial $\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{8}\mathit{x}\mathbf{+}\mathbf{4}$ by using the grouping method:

Find the product of the quadratic term $3x^2$ and the constant term $4$, $3x^2\times 4=12x^2$ .

By a hit and trial process, we have to find two factors that give this product $12x^2$, 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 $12x^2$.

Rewrite $8x$ as $6x+2x$ in the expression $3x^2+8x+4$.

Then factorize by grouping the terms.

$\begin{array}{rcl}3x^2+8x+4& =& 3x^2+6x+2x+4\\ & =& \left(3x^2+6x\right)+\left(2x+4\right)\\ & =& 3x\left(x+2\right)+2\left(x+2\right)\\ & =& \left(x+2\right)\left(3x+2\right)\end{array}$

Therefore, $3x^2+8x+4$ is factorized to $\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathbf{(}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{2}\mathbf{)}$ by using the grouping method as shown above.