Factor polynomial $$ 4x^{2}-12x-7 $$

$$ 4x^{2}-12x-7 = ( 2x-7 )( 2x+1 ) $$

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:

$$ a = 4 , b = -12 ~ \text{ and } ~ c = -7 $$

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 = -12 $.

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 $\color{blue}{ b = -12 }$

PRODUCT = -28 and SUM = -12
-1     28	1     -28
-2     14	2     -14
-4     7	4     -7

Step 6: Replace middle term $ -12 x $ with $ 2x-14x $:

$$ 4x^{2}-12x-7 = 4x^{2}+2x-14x-7 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 4x^{2}+2x-14x-7 = 2x\left(2x+1\right) -7\left(2x+1\right) = \left(2x-7\right) \left(2x+1\right) $$

Polynomial Factoring Calculator