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:

$$ a = 8 , b = -44 ~ \text{ and } ~ c = 48 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = 48} $.

$$ a \cdot c = 384 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 384 $ and add to $ b = -44 $.

Step 4: All pairs of numbers with a product of $ 384 $ are:

PRODUCT = 384
1     384	-1     -384
2     192	-2     -192
3     128	-3     -128
4     96	-4     -96
6     64	-6     -64
8     48	-8     -48
12     32	-12     -32
16     24	-16     -24

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -44 }$

PRODUCT = 384 and SUM = -44
1     384	-1     -384
2     192	-2     -192
3     128	-3     -128
4     96	-4     -96
6     64	-6     -64
8     48	-8     -48
12     32	-12     -32
16     24	-16     -24

Step 6: Replace middle term $ -44 x $ with $ -12x-32x $:

$$ 8x^{2}-44x+48 = 8x^{2}-12x-32x+48 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -16 $ out of the last two terms.

$$ 8x^{2}-12x-32x+48 = 4x\left(2x-3\right) -16\left(2x-3\right) = \left(4x-16\right) \left(2x-3\right) $$

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