To simplify the expression, let's first factorize the denominators of both fractions:
For the first fraction: (6-x)/(x^2 + 2x - 3) = (6-x)/[(x-1)(x+3)]
For the second fraction: (x^2 - 4x - 12)/(x^2 + 4x + 3) = (x^2 - 4x - 12)/[(x+3)(x+1)]
Now we rewrite the expression with the simplified fractions:
[(6-x)/[(x-1)(x+3)] + (x^2 - 4x - 12)/[(x+3)(x+1)]
To combine the fractions, we need to find a common denominator, which is (x-1)(x+1)(x+3). Rewrite each fraction with this common denominator:
(6-x)(x+1)/(x-1)(x+3)(x+1) + (x^2 - 4x - 12)(x-1)/(x+1)(x+3)(x-1)
Now we can combine the fractions into one using the common denominator:
[ (6-x)(x+1) + (x^2 - 4x - 12)(x-1) ] / [(x-1)(x+1)(x+3)]
Expand the numerator to simplify further:
[ (6x + 6 - x^2 - x) + (x^3 - x^2 - 4x^2 + 4x - 12x + 12) ] / [(x-1)(x+1)(x+3)]
Combine like terms in the numerator:
[ (5x + 6 - x^2 - x) + (x^3 - 5x^2 - 12x + 12) ] / [(x-1)(x+1)(x+3)]
Simplify it further:
[ x^3 - 6x^2 - 6x + 6 ] / [(x-1)(x+1)(x+3)]
Therefore, the simplified form of the quotient (6-x)/(x^2 + 2x - 3) + (x^2 - 4x - 12)/(x^2 + 4x + 3) is (x^3 - 6x^2 - 6x + 6)/[(x-1)(x+1)(x+3)] with the restrictions x ≠ -3, -1, 1.
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.
x not equal -3, -3, 1, 6
x not equal -3, -2, 6
x not equal -2, 1
x not equal -2
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