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:

$$ a = 4 , b = 7 ~ \text{ and } ~ c = -2 $$

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

$$ a \cdot c = -8 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -8 $ and add to $ b = 7 $.

Step 4: All pairs of numbers with a product of $ -8 $ are:

PRODUCT = -8
-1     8	1     -8
-2     4	2     -4

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 7 }$

PRODUCT = -8 and SUM = 7
-1     8	1     -8
-2     4	2     -4

Step 6: Replace middle term $ 7 x $ with $ 8x-x $:

$$ 4x^{2}+7x-2 = 4x^{2}+8x-x-2 $$

Step 7: Apply factoring by grouping. Factor $ 4x $ out of the first two terms and $ -1 $ out of the last two terms.

$$ 4x^{2}+8x-x-2 = 4x\left(x+2\right) -1\left(x+2\right) = \left(4x-1\right) \left(x+2\right) $$

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