What is the quotient in the simplest form? State any restrictions on the variable.
(x+2)/(x-1)/(x+4/x^2+4x-5)
To simplify the expression, we need to divide the fraction (x+2)/(x-1) by the fraction (x+4)/(x^2+4x-5).
First, let's simplify each fraction individually:
The fraction (x+2)/(x-1) can't be simplified any further.
The fraction (x+4)/(x^2+4x-5) can be simplified as follows:
(x+4)/(x^2+4x-5) = (x+4)/[(x+5)(x-1)]
Now, let's divide the two fractions by multiplying the first fraction by the reciprocal of the second fraction:
(x+2)/(x-1) * [(x+5)(x-1)/(x+4)]
= [(x+2)(x+5)(x-1)]/[(x-1)(x+4)]
= (x+2)(x+5)/(x+4)
The quotient in its simplest form is (x+2)(x+5)/(x+4).
As for restrictions on the variable, we need to consider any values that would make the denominator of the original fractions equal to zero and exclude them from the domain. In this case, we have x-1=0 and x^2+4x-5=0.
Solving x-1=0 gives x=1, and solving x^2+4x-5=0 gives x=-5 or x=1.
Therefore, the restrictions on the variable are x≠-5 and x≠1.