4y2-92y2+11y-21÷2y2+y-3y2+6y-7

To divide by a fraction, multiply by its reciprocal.

4y2-92y2+11y-21⋅y2+6y-72y2+y-3

Rewrite 4y2 as (2y)2.

(2y)2-92y2+11y-21⋅y2+6y-72y2+y-3

Rewrite 9 as 32.

(2y)2-322y2+11y-21⋅y2+6y-72y2+y-3

Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2y and b=3.

(2y+3)(2y-3)2y2+11y-21⋅y2+6y-72y2+y-3

(2y+3)(2y-3)2y2+11y-21⋅y2+6y-72y2+y-3

For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=2⋅-21=-42 and whose sum is b=11.

Factor 11 out of 11y.

(2y+3)(2y-3)2y2+11(y)-21⋅y2+6y-72y2+y-3

Rewrite 11 as -3 plus 14

(2y+3)(2y-3)2y2+(-3+14)y-21⋅y2+6y-72y2+y-3

Apply the distributive property.

(2y+3)(2y-3)2y2-3y+14y-21⋅y2+6y-72y2+y-3

(2y+3)(2y-3)2y2-3y+14y-21⋅y2+6y-72y2+y-3

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

(2y+3)(2y-3)(2y2-3y)+14y-21⋅y2+6y-72y2+y-3

Factor out the greatest common factor (GCF) from each group.

(2y+3)(2y-3)y(2y-3)+7(2y-3)⋅y2+6y-72y2+y-3

(2y+3)(2y-3)y(2y-3)+7(2y-3)⋅y2+6y-72y2+y-3

Factor the polynomial by factoring out the greatest common factor, 2y-3.

(2y+3)(2y-3)(2y-3)(y+7)⋅y2+6y-72y2+y-3

(2y+3)(2y-3)(2y-3)(y+7)⋅y2+6y-72y2+y-3

Cancel the common factor.

(2y+3)(2y-3)(2y-3)(y+7)⋅y2+6y-72y2+y-3

Rewrite the expression.

2y+3y+7⋅y2+6y-72y2+y-3

2y+3y+7⋅y2+6y-72y2+y-3

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 -7 and whose sum is 6.

-1,7

Write the factored form using these integers.

2y+3y+7⋅(y-1)(y+7)2y2+y-3

2y+3y+7⋅(y-1)(y+7)2y2+y-3

For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=2⋅-3=-6 and whose sum is b=1.

Multiply by 1.

2y+3y+7⋅(y-1)(y+7)2y2+1y-3

Rewrite 1 as -2 plus 3

2y+3y+7⋅(y-1)(y+7)2y2+(-2+3)y-3

Apply the distributive property.

2y+3y+7⋅(y-1)(y+7)2y2-2y+3y-3

2y+3y+7⋅(y-1)(y+7)2y2-2y+3y-3

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

2y+3y+7⋅(y-1)(y+7)(2y2-2y)+3y-3

Factor out the greatest common factor (GCF) from each group.

2y+3y+7⋅(y-1)(y+7)2y(y-1)+3(y-1)

2y+3y+7⋅(y-1)(y+7)2y(y-1)+3(y-1)

Factor the polynomial by factoring out the greatest common factor, y-1.

2y+3y+7⋅(y-1)(y+7)(y-1)(2y+3)

2y+3y+7⋅(y-1)(y+7)(y-1)(2y+3)

Factor 2y+3 out of (y-1)(2y+3).

2y+3y+7⋅(y-1)(y+7)(2y+3)(y-1)

Cancel the common factor.

2y+3y+7⋅(y-1)(y+7)(2y+3)(y-1)

Rewrite the expression.

1y+7⋅(y-1)(y+7)y-1

1y+7⋅(y-1)(y+7)y-1

Factor y+7 out of (y-1)(y+7).

1y+7⋅(y+7)(y-1)y-1

Cancel the common factor.

1y+7⋅(y+7)(y-1)y-1

Rewrite the expression.

y-1y-1

y-1y-1

Cancel the common factor.

y-1y-1

Divide 1 by 1.

1

1

Simplify ((4y^2-9)/(2y^2+11y-21))÷((2y^2+y-3)/(y^2+6y-7))