what is the quotient (6-x)/(x^2 + 2x - 3) + (x^2 - 4x - 12)/(x^2 + 4x + 3) in simplified form? state any restrictions on the variable.

Answer 1

To simplify the expression, we need to find a common denominator for the two fractions and then combine them into a single fraction.

First, factor the denominators:
x^2 + 2x - 3 = (x + 3)(x - 1)
x^2 + 4x + 3 = (x + 3)(x + 1)

So, the expression becomes:
((6 - x)/(x + 3)(x - 1)) + ((x^2 - 4x - 12)/(x + 3)(x + 1))

Now, find a common denominator:
(x + 3)(x - 1)(x + 1)

Next, rewrite each fraction with the common denominator:
( (6 - x)(x + 1) + (x^2 - 4x - 12)(x - 1) ) / (x + 3)(x - 1)(x + 1)

Simplify the numerator:
( 6x + 6 - x^2 - x + x^2 - 4x + 12 ) / (x + 3)(x - 1)(x + 1)
( 2 - 3x ) / (x + 3)(x - 1)(x + 1)

Therefore, the simplified form of the expression is (2 - 3x) / (x + 3)(x - 1)(x + 1)

Restrictions on the variable:
The variable x cannot be equal to -3, 1, or -1 since these values would result in division by zero in the denominator. So the restrictions are x ≠ -3, x ≠ 1, x ≠ -1.