Step 1: Identify constants $ a $ , $ b $ and $ c $.
$ a $ is a number in front of the $ x^2 $ term $ b $ is a number in front of the $ x $ term and $ c $ is a constant. In this case:
Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = 4} $.
$$ a \cdot c = 24 $$Step 3: Find out two numbers that multiply to $ a \cdot c = 24 $ and add to $ b = 11 $.
Step 4: All pairs of numbers with a product of $ 24 $ are:
PRODUCT = 24 | |
1 24 | -1 -24 |
2 12 | -2 -12 |
3 8 | -3 -8 |
4 6 | -4 -6 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = 11 }$
PRODUCT = 24 and SUM = 11 | |
1 24 | -1 -24 |
2 12 | -2 -12 |
3 8 | -3 -8 |
4 6 | -4 -6 |
Step 6: Replace middle term $ 11 x $ with $ 8x+3x $:
$$ 6x^{2}+11x+4 = 6x^{2}+8x+3x+4 $$Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ 1 $ out of the last two terms.
$$ 6x^{2}+8x+3x+4 = 2x\left(3x+4\right) + 1\left(3x+4\right) = \left(2x+1\right) \left(3x+4\right) $$