Step 1 :

After factoring out $ -3y $ we have:

$$ -24y^{3}+90y^{2}-75y = -3y ( 8y^{2}-30y+25 ) $$

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:

$$ a = 8 , b = -30 ~ \text{ and } ~ c = 25 $$

Step 3: Multiply the leading coefficient $\color{blue}{ a = 8 }$ by the constant term $\color{blue}{c = 25} $.

$$ a \cdot c = 200 $$

Step 4: Find out two numbers that multiply to $ a \cdot c = 200 $ and add to $ b = -30 $.

Step 5: All pairs of numbers with a product of $ 200 $ are:

PRODUCT = 200
1     200	-1     -200
2     100	-2     -100
4     50	-4     -50
5     40	-5     -40
8     25	-8     -25
10     20	-10     -20

Step 6: Find out which factor pair sums up to $\color{blue}{ b = -30 }$

PRODUCT = 200 and SUM = -30
1     200	-1     -200
2     100	-2     -100
4     50	-4     -50
5     40	-5     -40
8     25	-8     -25
10     20	-10     -20

Step 7: Replace middle term $ -30 x $ with $ -10x-20x $:

$$ 8x^{2}-30x+25 = 8x^{2}-10x-20x+25 $$

Step 8: Apply factoring by grouping. Factor $ 2x $ out of the first two terms and $ -5 $ out of the last two terms.

$$ 8x^{2}-10x-20x+25 = 2x\left(4x-5\right) -5\left(4x-5\right) = \left(2x-5\right) \left(4x-5\right) $$

Polynomial Factoring Calculator