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 $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = -7} $.
$$ a \cdot c = -28 $$Step 3: Find out two numbers that multiply to $ a \cdot c = -28 $ and add to $ b = -3 $.
Step 4: All pairs of numbers with a product of $ -28 $ are:
PRODUCT = -28 | |
-1 28 | 1 -28 |
-2 14 | 2 -14 |
-4 7 | 4 -7 |
Step 5: Find out which factor pair sums up to $\color{blue}{ b = -3 }$
PRODUCT = -28 and SUM = -3 | |
-1 28 | 1 -28 |
-2 14 | 2 -14 |
-4 7 | 4 -7 |
Step 6: Replace middle term $ -3 x $ with $ 4x-7x $:
$$ 4x^{2}-3x-7 = 4x^{2}+4x-7x-7 $$Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -7 $ out of the last two terms.
$$ 4x^{2}+4x-7x-7 = 4x\left(x+1\right) -7\left(x+1\right) = \left(4x-7\right) \left(x+1\right) $$