Factor polynomial $$ 4x^{2}-4x-3 $$

$$ 4x^{2}-4x-3 = ( 2x-3 )( 2x+1 ) $$

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:

$$ a = 4 , b = -4 ~ \text{ and } ~ c = -3 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = -3} $.

$$ a \cdot c = -12 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -12 $ and add to $ b = -4 $.

Step 4: All pairs of numbers with a product of $ -12 $ are:

PRODUCT = -12
-1     12	1     -12
-2     6	2     -6
-3     4	3     -4

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -4 }$

PRODUCT = -12 and SUM = -4
-1     12	1     -12
-2     6	2     -6
-3     4	3     -4

Step 6: Replace middle term $ -4 x $ with $ 2x-6x $:

$$ 4x^{2}-4x-3 = 4x^{2}+2x-6x-3 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 4x^{2}+2x-6x-3 = 2x\left(2x+1\right) -3\left(2x+1\right) = \left(2x-3\right) \left(2x+1\right) $$

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