Step 1 :

After factoring out $ 2 $ we have:

$$ 250x^{3}-16 = 2 ( 125x^{3}-8 ) $$

Step 2 :

To factor $ 125x^{3}-8 $ we can use difference of cubes formula:

$$ I^3 - II^3 = (I - II)(I^2 + I \cdot II + II^2) $$

After putting $ I = 5x $ and $ II = 2 $ , we have:

$$ 125x^{3}-8 = ( 5x-2 ) ( 25x^{2}+10x+4 ) $$

Step 3 :

Step 3: 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:

$$ a = 25 , b = 10 ~ \text{ and } ~ c = 4 $$

Step 4: Multiply the leading coefficient $\color{blue}{ a = 25 }$ by the constant term $\color{blue}{c = 4} $.

$$ a \cdot c = 100 $$

Step 5: Find out two numbers that multiply to $ a \cdot c = 100 $ and add to $ b = 10 $.

Step 6: All pairs of numbers with a product of $ 100 $ are:

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

Step 7: Find out which factor pair sums up to $\color{blue}{ b = 10 }$

Step 8: Because none of these pairs will give us a sum of $ \color{blue}{ 10 }$, we conclude the polynomial cannot be factored.

Polynomial Factoring Calculator