What is the product in simplest form? Show all steps
(x+2)(x−2)/(x+1)(x−1)⋅(x+1)/x(x+2)
To find the product of the given expression in simplest form, we need to first simplify each fraction and then multiply them together.
1. First, simplify each fraction separately:
(x+2)(x-2)/(x+1)(x-1) = [(x^2 - 4)/(x^2 - 1)] = [(x+2)(x-2)/(x+1)(x-1)]
(x+1)/x = 1 + 1/x
2. Now, multiply the two fractions together:
[(x+2)(x-2)/(x+1)(x-1)] * [(x+1)/x] = [(x+2)(x-2)(x+1)]/[(x+1)(x-1)x]
3. Simplify the expression by expanding and canceling out common terms:
[(x+2)(x-2)(x+1)]/[(x+1)(x-1)x]
= (x^2 - 4)(x+1)/(x^2 - 1)x
= (x^3 + x^2 - 4x - 4)/(x^3 - x)
= x(x^2 + x - 4 - 4/x)/(x^3 - x)
= (x^3 + x^2 - 4x - 4 - 4)/(x^3 - x)
= (x^3 + x^2 - 4x - 8)/(x^3 - x)
Therefore, the product of the given expression in simplest form is: (x^3 + x^2 - 4x - 8)/(x^3 - x)