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:

$$ a = 10 , b = -13 ~ \text{ and } ~ c = -30 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 10 }$ by the constant term $\color{blue}{c = -30} $.

$$ a \cdot c = -300 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -300 $ and add to $ b = -13 $.

Step 4: All pairs of numbers with a product of $ -300 $ are:

PRODUCT = -300
-1     300	1     -300
-2     150	2     -150
-3     100	3     -100
-4     75	4     -75
-5     60	5     -60
-6     50	6     -50
-10     30	10     -30
-12     25	12     -25
-15     20	15     -20

Step 5: Find out which factor pair sums up to $\color{blue}{ b = -13 }$

PRODUCT = -300 and SUM = -13
-1     300	1     -300
-2     150	2     -150
-3     100	3     -100
-4     75	4     -75
-5     60	5     -60
-6     50	6     -50
-10     30	10     -30
-12     25	12     -25
-15     20	15     -20

Step 6: Replace middle term $ -13 x $ with $ 12x-25x $:

$$ 10x^{2}-13x-30 = 10x^{2}+12x-25x-30 $$

Step 7: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 10x^{2}+12x-25x-30 = 2x\left(5x+6\right) -5\left(5x+6\right) = \left(2x-5\right) \left(5x+6\right) $$

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