What is the quotient 6-x/x^2+2x-3 divided by x^2-4x-12/x^2+4x+3 in simplified form?
We can start by factoring the denominators of both fractions:
6 - x = (3 - x)(2 + x)
x^2 + 2x - 3 = (x - 1)(x + 3)
x^2 - 4x - 12 = (x - 6)(x + 2)
x^2 + 4x + 3 = (x + 3)(x + 1)
Now we can rewrite the quotient as a multiplication by flipping the second fraction and simplifying:
(6 - x)/(x^2 + 2x - 3) ÷ (x^2 - 4x - 12)/(x^2 + 4x + 3) = (3 - x)(2 + x)/(x - 1)(x + 3) * (x + 4)(x - 3)/(x - 6)(x + 2)
Next, we can cancel out any factors that appear in both the numerator and denominator:
(3 - x)(2 + x)/(x - 1)(x + 3) * (x + 4)(x - 3)/(x - 6)(x + 2) = -(x - 3)(x + 4)/(x - 1)(x + 3) * (x + 4)(x - 3)/(x - 6)(x + 2)
Finally, we can multiply the remaining factors together and simplify:
-(x - 3)(x + 4)/(x - 1)(x + 3) * (x + 4)(x - 3)/(x - 6)(x + 2) = -1/(x - 1)(x + 2)