Step 1 :

After factoring out $ -1 $ we have:

$$ -4x^{2}+4x+15 = - ~ ( 4x^{2}-4x-15 ) $$

Step 2 :

Step 2: 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:

$$ a = 4 , b = -4 ~ \text{ and } ~ c = -15 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = -15} $.

$$ a \cdot c = -60 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -60 $ and add to $ b = -4 $.

Step 5: All pairs of numbers with a product of $ -60 $ are:

PRODUCT = -60
-1     60	1     -60
-2     30	2     -30
-3     20	3     -20
-4     15	4     -15
-5     12	5     -12
-6     10	6     -10

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -4 }$

PRODUCT = -60 and SUM = -4
-1     60	1     -60
-2     30	2     -30
-3     20	3     -20
-4     15	4     -15
-5     12	5     -12
-6     10	6     -10

Step 7: Replace middle term $ -4 x $ with $ 6x-10x $:

$$ 4x^{2}-4x-15 = 4x^{2}+6x-10x-15 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 4x^{2}+6x-10x-15 = 2x\left(2x+3\right) -5\left(2x+3\right) = \left(2x-5\right) \left(2x+3\right) $$

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