12n2-3632n2-25n+77÷14n2+73n-22n2-15n+56

To divide by a fraction, multiply by its reciprocal.

12n2-3632n2-25n+77⋅n2-15n+5614n2+73n-22

Factor 3 out of 12n2-363.

Factor 3 out of 12n2.

3(4n2)-3632n2-25n+77⋅n2-15n+5614n2+73n-22

Factor 3 out of -363.

3(4n2)+3(-121)2n2-25n+77⋅n2-15n+5614n2+73n-22

Factor 3 out of 3(4n2)+3(-121).

3(4n2-121)2n2-25n+77⋅n2-15n+5614n2+73n-22

3(4n2-121)2n2-25n+77⋅n2-15n+5614n2+73n-22

Rewrite 4n2 as (2n)2.

3((2n)2-121)2n2-25n+77⋅n2-15n+5614n2+73n-22

Rewrite 121 as 112.

3((2n)2-112)2n2-25n+77⋅n2-15n+5614n2+73n-22

Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=2n and b=11.

3(2n+11)(2n-11)2n2-25n+77⋅n2-15n+5614n2+73n-22

3(2n+11)(2n-11)2n2-25n+77⋅n2-15n+5614n2+73n-22

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⋅77=154 and whose sum is b=-25.

Factor -25 out of -25n.

3(2n+11)(2n-11)2n2-25(n)+77⋅n2-15n+5614n2+73n-22

Rewrite -25 as -11 plus -14

3(2n+11)(2n-11)2n2+(-11-14)n+77⋅n2-15n+5614n2+73n-22

Apply the distributive property.

3(2n+11)(2n-11)2n2-11n-14n+77⋅n2-15n+5614n2+73n-22

3(2n+11)(2n-11)2n2-11n-14n+77⋅n2-15n+5614n2+73n-22

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

3(2n+11)(2n-11)(2n2-11n)-14n+77⋅n2-15n+5614n2+73n-22

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

3(2n+11)(2n-11)n(2n-11)-7(2n-11)⋅n2-15n+5614n2+73n-22

3(2n+11)(2n-11)n(2n-11)-7(2n-11)⋅n2-15n+5614n2+73n-22

Factor the polynomial by factoring out the greatest common factor, 2n-11.

3(2n+11)(2n-11)(2n-11)(n-7)⋅n2-15n+5614n2+73n-22

3(2n+11)(2n-11)(2n-11)(n-7)⋅n2-15n+5614n2+73n-22

Cancel the common factor.

3(2n+11)(2n-11)(2n-11)(n-7)⋅n2-15n+5614n2+73n-22

Rewrite the expression.

3(2n+11)n-7⋅n2-15n+5614n2+73n-22

3(2n+11)n-7⋅n2-15n+5614n2+73n-22

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 56 and whose sum is -15.

-8,-7

Write the factored form using these integers.

3(2n+11)n-7⋅(n-8)(n-7)14n2+73n-22

3(2n+11)n-7⋅(n-8)(n-7)14n2+73n-22

For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=14⋅-22=-308 and whose sum is b=73.

Factor 73 out of 73n.

3(2n+11)n-7⋅(n-8)(n-7)14n2+73(n)-22

Rewrite 73 as -4 plus 77

3(2n+11)n-7⋅(n-8)(n-7)14n2+(-4+77)n-22

Apply the distributive property.

3(2n+11)n-7⋅(n-8)(n-7)14n2-4n+77n-22

3(2n+11)n-7⋅(n-8)(n-7)14n2-4n+77n-22

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

3(2n+11)n-7⋅(n-8)(n-7)(14n2-4n)+77n-22

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

3(2n+11)n-7⋅(n-8)(n-7)2n(7n-2)+11(7n-2)

3(2n+11)n-7⋅(n-8)(n-7)2n(7n-2)+11(7n-2)

Factor the polynomial by factoring out the greatest common factor, 7n-2.

3(2n+11)n-7⋅(n-8)(n-7)(7n-2)(2n+11)

3(2n+11)n-7⋅(n-8)(n-7)(7n-2)(2n+11)

Factor 2n+11 out of 3(2n+11).

(2n+11)⋅3n-7⋅(n-8)(n-7)(7n-2)(2n+11)

Factor 2n+11 out of (7n-2)(2n+11).

(2n+11)⋅3n-7⋅(n-8)(n-7)(2n+11)(7n-2)

Cancel the common factor.

(2n+11)⋅3n-7⋅(n-8)(n-7)(2n+11)(7n-2)

Rewrite the expression.

3n-7⋅(n-8)(n-7)7n-2

3n-7⋅(n-8)(n-7)7n-2

Factor n-7 out of (n-8)(n-7).

3n-7⋅(n-7)(n-8)7n-2

Cancel the common factor.

3n-7⋅(n-7)(n-8)7n-2

Rewrite the expression.

3n-87n-2

3n-87n-2

Combine 3 and n-87n-2.

3(n-8)7n-2

Simplify ((12n^2-363)/(2n^2-25n+77))÷((14n^2+73n-22)/(n^2-15n+56))