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 first need to find a common denominator for the two fractions:

The denominator of the first fraction, (x^2 + 2x - 3), factors into (x + 3)(x - 1).
The denominator of the second fraction, (x^2 + 4x + 3), factors into (x + 3)(x + 1).

So, the common denominator is (x + 3)(x - 1)(x + 1).

Now, we rewrite the fractions with the common denominator and add them:

[(6 - x)(x + 1) + (x^2 - 4x - 12)(x - 1)] / [(x + 3)(x - 1)(x + 1)]

Expanding the numerators:

[(6x + 6 - x^2 - x) + (x^3 - x^2 - 4x^2 + 4x - 12x + 12)] / [(x + 3)(x - 1)(x + 1)]

Combining like terms in the numerator:

(5x + 6 - x^2 - x + x^3 - x^2 - 4x^2 - 8x + 12) / [(x + 3)(x - 1)(x + 1)]

Simplifying further:

(x^3 - 6x^2 - 4x + 18) / [(x + 3)(x - 1)(x + 1)]

So, the quotient of [(6-x)/(x^2+2x-3)] + [(x^2-4x-12)/(x^2+4x+3)] simplifies to (x^3 - 6x^2 - 4x + 18) / [(x + 3)(x - 1)(x + 1)].

Restrictions on the variable:
The variables x cannot be equal to -3, 1, or -1 because those values would make the denominator equal to zero, resulting in undefined expressions.