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

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We have to factorize 36 - 12(y+y^2) + (y+y^2)^2

36 - 12(y+y^2) + (y+y^2)^2

=> 6^2 - 2*6*(y + y^2) + (y + y^2)^2

=> [6 - (y + y^2)]^2

=> [y^2 + y - 6]^2

=> [y^2 + 3y - 2y - 6]^2

=> [y(y + 3) - 2(y + 3)]^2

=> [(y - 2)(y + 3)]^2

**The given expression 36 - 12(y+y^2) + (y+y^2)^2 = [(y - 2)(y + 3)]^2**

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**