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:

$$ a = 16 , b = -8 ~ \text{ and } ~ c = -63 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 16 }$ by the constant term $\color{blue}{c = -63} $.

$$ a \cdot c = -1008 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -1008 $ and add to $ b = -8 $.

Step 4: All pairs of numbers with a product of $ -1008 $ are:

PRODUCT = -1008
-1     1008	1     -1008
-2     504	2     -504
-3     336	3     -336
-4     252	4     -252
-6     168	6     -168
-7     144	7     -144
-8     126	8     -126
-9     112	9     -112
-12     84	12     -84
-14     72	14     -72
-16     63	16     -63
-18     56	18     -56
-21     48	21     -48
-24     42	24     -42
-28     36	28     -36

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -8 }$

PRODUCT = -1008 and SUM = -8
-1     1008	1     -1008
-2     504	2     -504
-3     336	3     -336
-4     252	4     -252
-6     168	6     -168
-7     144	7     -144
-8     126	8     -126
-9     112	9     -112
-12     84	12     -84
-14     72	14     -72
-16     63	16     -63
-18     56	18     -56
-21     48	21     -48
-24     42	24     -42
-28     36	28     -36

Step 6: Replace middle term $ -8 x $ with $ 28x-36x $:

$$ 16x^{2}-8x-63 = 16x^{2}+28x-36x-63 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -9 $ out of the last two terms.

$$ 16x^{2}+28x-36x-63 = 4x\left(4x+7\right) -9\left(4x+7\right) = \left(4x-9\right) \left(4x+7\right) $$

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