y2-6y+9-(y2-4)÷y2-9y2-8y+12

To divide by a fraction, multiply by its reciprocal.

y2-6y+9-(y2-4)⋅y2-8y+12y2-9

Rewrite 9 as 32.

y2-6y+32-(y2-4)⋅y2-8y+12y2-9

Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.

2ab=2⋅y⋅-3

Simplify.

2ab=-6y

Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=y and b=-3.

(y-3)2-(y2-4)⋅y2-8y+12y2-9

(y-3)2-(y2-4)⋅y2-8y+12y2-9

Rewrite 4 as 22.

(y-3)2-(y2-22)⋅y2-8y+12y2-9

Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=y and b=2.

(y-3)2-(y+2)(y-2)⋅y2-8y+12y2-9

(y-3)2-(y+2)(y-2)⋅y2-8y+12y2-9

Move the negative in front of the fraction.

-(y-3)2(y+2)(y-2)⋅y2-8y+12y2-9

Consider the form x2+bx+c. Find a pair of integers whose product is c and whose sum is b. In this case, whose product is 12 and whose sum is -8.

-6,-2

Write the factored form using these integers.

-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)y2-9

-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)y2-9

Rewrite 9 as 32.

-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)y2-32

Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=y and b=3.

-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)

-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)

Cancel the common factor of y-3.

Move the leading negative in -(y-3)2(y+2)(y-2) into the numerator.

-(y-3)2(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)

Factor y-3 out of -(y-3)2.

(y-3)(-(y-3))(y+2)(y-2)⋅(y-6)(y-2)(y+3)(y-3)

Factor y-3 out of (y+3)(y-3).

(y-3)(-(y-3))(y+2)(y-2)⋅(y-6)(y-2)(y-3)(y+3)

Cancel the common factor.

(y-3)(-(y-3))(y+2)(y-2)⋅(y-6)(y-2)(y-3)(y+3)

Rewrite the expression.

-(y-3)(y+2)(y-2)⋅(y-6)(y-2)y+3

-(y-3)(y+2)(y-2)⋅(y-6)(y-2)y+3

Cancel the common factor of y-2.

Factor y-2 out of (y+2)(y-2).

-(y-3)(y-2)(y+2)⋅(y-6)(y-2)y+3

Factor y-2 out of (y-6)(y-2).

-(y-3)(y-2)(y+2)⋅(y-2)(y-6)y+3

Cancel the common factor.

-(y-3)(y-2)(y+2)⋅(y-2)(y-6)y+3

Rewrite the expression.

-(y-3)y+2⋅y-6y+3

-(y-3)y+2⋅y-6y+3

Multiply -(y-3)y+2 and y-6y+3.

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

Move the negative in front of the fraction.

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

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

Simplify ((y^2-6y+9)/(-(y^2-4)))÷((y^2-9)/(y^2-8y+12))