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 $\color{blue}{ a = 48 }$ by the constant term $\color{blue}{c = -15} $.
$$ a \cdot c = -720 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -720 $ and add to $ b = 22 $.
Step 4: All pairs of numbers with a product of $ -720 $ are:
PRODUCT = -720 | |
-1 720 | 1 -720 |
-2 360 | 2 -360 |
-3 240 | 3 -240 |
-4 180 | 4 -180 |
-5 144 | 5 -144 |
-6 120 | 6 -120 |
-8 90 | 8 -90 |
-9 80 | 9 -80 |
-10 72 | 10 -72 |
-12 60 | 12 -60 |
-15 48 | 15 -48 |
-16 45 | 16 -45 |
-18 40 | 18 -40 |
-20 36 | 20 -36 |
-24 30 | 24 -30 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = 22 }$
PRODUCT = -720 and SUM = 22 | |
-1 720 | 1 -720 |
-2 360 | 2 -360 |
-3 240 | 3 -240 |
-4 180 | 4 -180 |
-5 144 | 5 -144 |
-6 120 | 6 -120 |
-8 90 | 8 -90 |
-9 80 | 9 -80 |
-10 72 | 10 -72 |
-12 60 | 12 -60 |
-15 48 | 15 -48 |
-16 45 | 16 -45 |
-18 40 | 18 -40 |
-20 36 | 20 -36 |
-24 30 | 24 -30 |
Step 6: Replace middle term $ 22 x $ with $ 40x-18x $:
$$ 48x^{2}+22x-15 = 48x^{2}+40x-18x-15 $$Step 7: Apply factoring by grouping. Factor $ 8x $ out of the first two terms and $ -3 $ out of the last two terms.
$$ 48x^{2}+40x-18x-15 = 8x\left(6x+5\right) -3\left(6x+5\right) = \left(8x-3\right) \left(6x+5\right) $$