What is the quotient 6-x/x^2+2x-3 divided by x^2-4x-12/x^2+4x+3 in simplified form? State any restrictions on the variable.
To divide these two rational expressions, we need to first factor all the polynomials involved:
6 - x = -(x - 6)
x^2 + 2x - 3 = (x + 3)(x - 1)
x^2 - 4x - 12 = (x - 6)(x + 2)
x^2 + 4x + 3 = (x + 1)(x + 3)
Now we can rewrite the division problem as multiplication by the reciprocal of the second fraction:
(6 - x)/(x^2 + 2x - 3) * (x^2 + 4x + 3)/(x^2 - 4x - 12)
Next, we can simplify the individual fractions by canceling factors:
(6 - x)/(x + 3)(x - 1) * (x + 1)(x + 3)/(x - 6)(x + 2)
Then, we can cancel out common factors in the numerator and denominator:
-(x - 6)(x + 1)/(x - 1)(x + 2)
This is the simplified form of the quotient. However, we need to consider any restrictions on the variable. From the factorization above, we see that the expressions in the denominators cannot equal zero. Therefore, we must exclude x = -3, 1, -2, and 6 from the domain of the quotient function.