# Simplify ((z^2-4)/(z-3))÷((z+2)/(z^2+z-12))

z2-4z-3÷z+2z2+z-12
To divide by a fraction, multiply by its reciprocal.
z2-4z-3⋅z2+z-12z+2
Simplify the numerator.
Rewrite 4 as 22.
z2-22z-3⋅z2+z-12z+2
Since both terms are perfect squares, factor using the difference of squares formula, a2-b2=(a+b)(a-b) where a=z and b=2.
(z+2)(z-2)z-3⋅z2+z-12z+2
(z+2)(z-2)z-3⋅z2+z-12z+2
Simplify terms.
Cancel the common factor of z+2.
Cancel the common factor.
(z+2)(z-2)z-3⋅z2+z-12z+2
Rewrite the expression.
z-2z-3(z2+z-12)
z-2z-3(z2+z-12)
Multiply z-2z-3 and z2+z-12.
(z-2)(z2+z-12)z-3
(z-2)(z2+z-12)z-3
Factor z2+z-12 using the AC method.
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 1.
-3,4
Write the factored form using these integers.
(z-2)((z-3)(z+4))z-3
(z-2)(z-3)(z+4)z-3
Cancel the common factor of z-3.
Cancel the common factor.
(z-2)(z-3)(z+4)z-3
Divide (z-2)(z+4) by 1.
(z-2)(z+4)
(z-2)(z+4)
Expand (z-2)(z+4) using the FOIL Method.
Apply the distributive property.
z(z+4)-2(z+4)
Apply the distributive property.
z⋅z+z⋅4-2(z+4)
Apply the distributive property.
z⋅z+z⋅4-2z-2⋅4
z⋅z+z⋅4-2z-2⋅4
Simplify and combine like terms.
Simplify each term.
Multiply z by z.
z2+z⋅4-2z-2⋅4
Move 4 to the left of z.
z2+4⋅z-2z-2⋅4
Multiply -2 by 4.
z2+4z-2z-8
z2+4z-2z-8
Subtract 2z from 4z.
z2+2z-8
z2+2z-8
Simplify ((z^2-4)/(z-3))÷((z+2)/(z^2+z-12))

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