36y⁴(y + 12)³ + y⁵(y + 12)⁴

Let (y + 12) = u  

Now rewrite the expression:

36y⁴u^{3 +} y⁵u⁴     GCF: y⁴u³  

y⁴u³(36 + yu)  

Now, for the sake of simplicity, revert u=(y+12) inside the parenthesis only:

y⁴u³(36 + y(y + 12))   

Now, expand the expression inside the parenthesis:

y⁴u³(36 + y² + 12y)

Now, rearrange the expression inside the parenthesis:

y⁴u³(y² + 12y + 36)

Look at that expression inside the parenthesis... is it a perfect square?  Indeed it is!!  36 = 6² and 12y = 2(6)y.  Then, factor it:

y⁴u³(y + 6)²

Finally... do you remember u = (y+12)? Now it's the time to revert the other u:

y⁴(y + 12)³(y + 6)²   Too much fun, isn't it?

                  ************************************

x² - 10x + 24

This trinomial may be factored as the product of two binomials if we find two numbers such that:

- Their product is equal to 24, and

- Their sum is equal to -10

These two numbers indeed exist:

they are -6 and -4:  (-6)(-4) = 24; and (-6)+(-4) = -10

Then our trinomial factors neatly: (x - 6)(x - 4)

Answer: (x - 6) (x - 4)

Because,

-6 * -4 = 24

-6x + -4x = -10x

x * x = x^2