Step 1 :

After factoring out $ 5 $ we have:

$$ 125y^{2}-50y-15 = 5 ( 25y^{2}-10y-3 ) $$

Step 2 :

Step 2: 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:

$$ a = 25 , b = -10 ~ \text{ and } ~ c = -3 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 25 }$ by the constant term $\color{blue}{c = -3} $.

$$ a \cdot c = -75 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = -75 $ and add to $ b = -10 $.

Step 5: All pairs of numbers with a product of $ -75 $ are:

PRODUCT = -75
-1     75	1     -75
-3     25	3     -25
-5     15	5     -15

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -10 }$

PRODUCT = -75 and SUM = -10
-1     75	1     -75
-3     25	3     -25
-5     15	5     -15

Step 7: Replace middle term $ -10 x $ with $ 5x-15x $:

$$ 25x^{2}-10x-3 = 25x^{2}+5x-15x-3 $$

Step 8: Apply factoring by grouping. Factor $ 5x $ out of the first two terms and $ -3 $ out of the last two terms.

$$ 25x^{2}+5x-15x-3 = 5x\left(5x+1\right) -3\left(5x+1\right) = \left(5x-3\right) \left(5x+1\right) $$

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