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:

$$ a = 7 , b = 8 ~ \text{ and } ~ c = 1 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 7 }$ by the constant term $\color{blue}{c = 1} $.

$$ a \cdot c = 7 $$

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

Step 4: All pairs of numbers with a product of $ 7 $ are:

PRODUCT = 7
1     7	-1     -7

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

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

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

$$ 7x^{2}+8x+1 = 7x^{2}+7x+x+1 $$

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

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

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