Factor Completely: 36-12(y+y^2)+(y+y^2)^2Show complete solution to explain the answer.

We notice that the given relation represents a complete
square. We'll use the special products to factor the
expression.

We'll recall that a binomial raised to square
is:

(a-b)^2 = a^2 - 2ab +
b^2

We'll identify a and b:

a
= 6 and b = -(y+y^2)

2ab =
-12(y+y^2)

Therefore, we can re-write the given expression
as:

E(x) = 36-12(y+y^2)+(y+y^2)^2 = (6 - y -
y^2)^2

We'll verify if the quadratic within brackets has
real roots.

y^2 + y - 6 = 0

y1
= [-1+sqrt(1+24)]/2

y1 =
(-1+5)/2

y1 = 2

y2 =
(-1-5)/2

y2 = -6/2

y2 =
-3

Therefore, the quadratic can be written as a product of
linear factors:

y^2 + y - 6 =
(y-y1)(y-y2)

y^2 + y - 6 =
(y-2)(y+3)

Therefore, the complete factorized
expression is: E(x)=[(y-2)(y+3)]^2