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:

$$ a = 7 , b = 48 ~ \text{ and } ~ c = -7 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = -7} $.

$$ a \cdot c = -49 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -49 $ and add to $ b = 48 $.

Step 4: All pairs of numbers with a product of $ -49 $ are:

PRODUCT = -49
-1     49	1     -49
-7     7	7     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = 48 }$

PRODUCT = -49 and SUM = 48
-1     49	1     -49
-7     7	7     -7

Step 6: Replace middle term $ 48 x $ with $ 49x-x $:

$$ 7x^{2}+48x-7 = 7x^{2}+49x-x-7 $$

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

$$ 7x^{2}+49x-x-7 = 7x\left(x+7\right) -1\left(x+7\right) = \left(7x-1\right) \left(x+7\right) $$

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