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:

$$ a = 4 , b = 3 ~ \text{ and } ~ c = -22 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 4 }$ by the constant term $\color{blue}{c = -22} $.

$$ a \cdot c = -88 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -88 $ and add to $ b = 3 $.

Step 4: All pairs of numbers with a product of $ -88 $ are:

PRODUCT = -88
-1     88	1     -88
-2     44	2     -44
-4     22	4     -22
-8     11	8     -11

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 3 }$

PRODUCT = -88 and SUM = 3
-1     88	1     -88
-2     44	2     -44
-4     22	4     -22
-8     11	8     -11

Step 6: Replace middle term $ 3 x $ with $ 11x-8x $:

$$ 4x^{2}+3x-22 = 4x^{2}+11x-8x-22 $$

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

$$ 4x^{2}+11x-8x-22 = x\left(4x+11\right) -2\left(4x+11\right) = \left(x-2\right) \left(4x+11\right) $$

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