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:

$$ a = 6 , b = 5 ~ \text{ and } ~ c = -25 $$

Step 2: Multiply the leading coefficient $\color{blue}{ a = 6 }$ by the constant term $\color{blue}{c = -25} $.

$$ a \cdot c = -150 $$

Step 3: Find out two numbers that multiply to $ a \cdot c = -150 $ and add to $ b = 5 $.

Step 4: All pairs of numbers with a product of $ -150 $ are:

PRODUCT = -150
-1     150	1     -150
-2     75	2     -75
-3     50	3     -50
-5     30	5     -30
-6     25	6     -25
-10     15	10     -15

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

PRODUCT = -150 and SUM = 5
-1     150	1     -150
-2     75	2     -75
-3     50	3     -50
-5     30	5     -30
-6     25	6     -25
-10     15	10     -15

Step 6: Replace middle term $ 5 x $ with $ 15x-10x $:

$$ 6x^{2}+5x-25 = 6x^{2}+15x-10x-25 $$

Step 7: Apply factoring by grouping. Factor $ 3x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 6x^{2}+15x-10x-25 = 3x\left(2x+5\right) -5\left(2x+5\right) = \left(3x-5\right) \left(2x+5\right) $$

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