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)
First, we need to factor the denominators of each expression:
(x^2 + 5x + 6) = (x + 2)(x + 3)
(x^2 - 2x - 8) = (x - 4)(x + 2)
Now rewrite the original expression with the factored denominators:
(x^2 - 16)/(x^2 + 5x + 6) + (x^2 + 5x + 4)/(x^2 - 2x - 8)
= (x^2 - 16)/[(x + 2)(x + 3)] + (x^2 + 5x + 4)/[(x - 4)(x + 2)]
Now find common denominators and combine the fractions:
[(x^2 - 16)(x - 4)]/[(x + 2)(x + 3)(x - 4)] + [(x^2 + 5x + 4)(x + 3)]/[(x - 4)(x + 2)(x + 3)]
= [(x - 4)(x + 4)]/[(x + 2)(x + 3)(x - 4)] + [(x + 1)(x + 4)]/[(x - 4)(x + 2)(x + 3)]
= [(x + 4)(x - 1)]/[(x + 2)(x + 3)(x - 4)]
Therefore, the simplified form of the expression is (x + 4)(x - 1)/[(x + 2)(x + 3)(x - 4)]. The only restriction on the variable is x cannot be equal to -2, -3, or 4 since these values would make the denominators equal to zero.