# Divide ((z^2+7+12)/(z^2+11z+24))÷((z^2+4z)/(z^2+6z-16))

z2+7+12z2+11z+24÷z2+4zz2+6z-16
To divide by a fraction, multiply by its reciprocal.
z2+7+12z2+11z+24⋅z2+6z-16z2+4z
z2+19z2+11z+24⋅z2+6z-16z2+4z
Factor z2+11z+24 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 24 and whose sum is 11.
3,8
Write the factored form using these integers.
z2+19(z+3)(z+8)⋅z2+6z-16z2+4z
z2+19(z+3)(z+8)⋅z2+6z-16z2+4z
Factor z2+6z-16 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 -16 and whose sum is 6.
-2,8
Write the factored form using these integers.
z2+19(z+3)(z+8)⋅(z-2)(z+8)z2+4z
z2+19(z+3)(z+8)⋅(z-2)(z+8)z2+4z
Factor z out of z2+4z.
Factor z out of z2.
z2+19(z+3)(z+8)⋅(z-2)(z+8)z⋅z+4z
Factor z out of 4z.
z2+19(z+3)(z+8)⋅(z-2)(z+8)z⋅z+z⋅4
Factor z out of z⋅z+z⋅4.
z2+19(z+3)(z+8)⋅(z-2)(z+8)z(z+4)
z2+19(z+3)(z+8)⋅(z-2)(z+8)z(z+4)
Cancel the common factor of z+8.
Factor z+8 out of (z+3)(z+8).
z2+19(z+8)(z+3)⋅(z-2)(z+8)z(z+4)
Factor z+8 out of (z-2)(z+8).
z2+19(z+8)(z+3)⋅(z+8)(z-2)z(z+4)
Cancel the common factor.
z2+19(z+8)(z+3)⋅(z+8)(z-2)z(z+4)
Rewrite the expression.
z2+19z+3⋅z-2z(z+4)
z2+19z+3⋅z-2z(z+4)
Multiply z2+19z+3 and z-2z(z+4).
(z2+19)(z-2)(z+3)(z(z+4))
Reorder factors in (z2+19)(z-2)(z+3)z(z+4).
(z2+19)(z-2)z(z+3)(z+4)
Divide ((z^2+7+12)/(z^2+11z+24))÷((z^2+4z)/(z^2+6z-16))