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## Factor The expression completely

36y4(y + 12)3 + y5(y + 12)4

#1: Follow the same pattern: 2 numbers such that their product is -30 and their sum is 1.  They're -6 and 5. Then: (x-6)(x+5)

#2: (9x+4)(3x+8)

#3: Difference of 2 squares: (6t+5s)(6t-5s)

#4: Factor by CF: 6x(x2 - 16) Now difference of 2 squares: 6x(x-4)(x+4)

#5: Same as #4: x6y5(x- y2); then x6y5(x - y)(x + y)

36y4(y + 12)3 + y5(y + 12)4

Let (y + 12) = u

Now rewrite the expression:

36y4u3 + 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 = 6and 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?

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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