what is the quotient in simplified form. state any restrictions on the variable.
x^2 - 16/x^2 + 5x + 6 + x^2 + 5x + 4/x^2 - 2x - 8
To simplify the given expression, we first factor the denominators and then combine the terms in the numerators:
(x^2 - 16) / (x^2 + 5x + 6) + (x^2 + 5x + 4) / (x^2 - 2x - 8)
(x^2 - 16) = (x + 4)(x - 4)
(x^2 + 5x + 6) = (x + 2)(x + 3)
(x^2 + 5x + 4) = (x + 4)(x + 1)
(x^2 - 2x - 8) = (x - 4)(x + 2)
Then the expression becomes:
[(x + 4)(x - 4) / (x + 2)(x + 3)] + [(x + 4)(x + 1) / (x - 4)(x + 2)]
Now, find the common denominator:
LCM of (x + 2)(x + 3) and (x - 4)(x + 2) = (x + 2)(x + 3)(x - 4)
Using the common denominator, we rewrite the expressions:
((x + 4)(x - 4) * (x - 4)) / ((x + 2)(x + 3)(x - 4)) + ((x + 4)(x + 1) * (x + 3)) / ((x + 2)(x + 3)(x - 4))
With a common denominator, we add the fractions:
[(x^2 - 16)(x - 4) + (x + 4)(x + 1)(x + 3)] / [(x + 2)(x + 3)(x - 4)]
Simplify further:
[(x^3 - 4x^2 - 16x + 64) + (x^3 + 8x^2 + 15x + 12)] / [(x + 2)(x + 3)(x - 4)]
Combine like terms:
2x^3 + 4x + 76 / [(x + 2)(x + 3)(x - 4)]
Therefore, the quotient in simplified form is:
(2x^3 + 4x + 76) / [(x + 2)(x + 3)(x - 4)]
There are no restrictions on the variable in this expression.