# Simplify ((z^2-4)/(z^2+2z-35))÷((z^2+2z)/(z^2-10z+25))

z2-4z2+2z-35÷z2+2zz2-10z+25
To divide by a fraction, multiply by its reciprocal.
z2-4z2+2z-35⋅z2-10z+25z2+2z
Simplify the numerator.
Rewrite 4 as 22.
z2-22z2+2z-35⋅z2-10z+25z2+2z
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)z2+2z-35⋅z2-10z+25z2+2z
(z+2)(z-2)z2+2z-35⋅z2-10z+25z2+2z
Factor z2+2z-35 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 -35 and whose sum is 2.
-5,7
Write the factored form using these integers.
(z+2)(z-2)(z-5)(z+7)⋅z2-10z+25z2+2z
(z+2)(z-2)(z-5)(z+7)⋅z2-10z+25z2+2z
Factor using the perfect square rule.
Rewrite 25 as 52.
(z+2)(z-2)(z-5)(z+7)⋅z2-10z+52z2+2z
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅z⋅-5
Simplify.
2ab=-10z
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=z and b=-5.
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2z2+2z
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2z2+2z
Simplify terms.
Factor z out of z2+2z.
Factor z out of z2.
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2z⋅z+2z
Factor z out of 2z.
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2z⋅z+z⋅2
Factor z out of z⋅z+z⋅2.
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2z(z+2)
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2z(z+2)
Cancel the common factor of z+2.
Factor z+2 out of z(z+2).
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2(z+2)z
Cancel the common factor.
(z+2)(z-2)(z-5)(z+7)⋅(z-5)2(z+2)z
Rewrite the expression.
z-2(z-5)(z+7)⋅(z-5)2z
z-2(z-5)(z+7)⋅(z-5)2z
Cancel the common factor of z-5.
Factor z-5 out of (z-5)2.
z-2(z-5)(z+7)⋅(z-5)(z-5)z
Cancel the common factor.
z-2(z-5)(z+7)⋅(z-5)(z-5)z
Rewrite the expression.
z-2z+7⋅z-5z
z-2z+7⋅z-5z
Multiply z-2z+7 and z-5z.
(z-2)(z-5)(z+7)z
Reorder factors in (z-2)(z-5)(z+7)z.
(z-2)(z-5)z(z+7)
(z-2)(z-5)z(z+7)
Simplify ((z^2-4)/(z^2+2z-35))÷((z^2+2z)/(z^2-10z+25))

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