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:

$$ a = 5 , b = -12 ~ \text{ and } ~ c = 7 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 5 }$ by the constant term $\color{blue}{c = 7} $.

$$ a \cdot c = 35 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = 35 $ and add to $ b = -12 $.

Step 4: All pairs of numbers with a product of $ 35 $ are:

PRODUCT = 35
1     35	-1     -35
5     7	-5     -7

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -12 }$

PRODUCT = 35 and SUM = -12
1     35	-1     -35
5     7	-5     -7

Step 6: Replace middle term $ -12 x $ with $ -5x-7x $:

$$ 5x^{2}-12x+7 = 5x^{2}-5x-7x+7 $$

Step 7: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -7 $ out of the last two terms.

$$ 5x^{2}-5x-7x+7 = 5x\left(x-1\right) -7\left(x-1\right) = \left(5x-7\right) \left(x-1\right) $$

Polynomial Factoring Calculator