Final answer:
The domain of the function f(x) = (x² + 2x - 8)/(x² + 6x + 8) is all real numbers except x = -4 and x = -2, which are the roots of the denominator when set equal to zero.
Step-by-step explanation:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) = (x² + 2x - 8)/(x² + 6x + 8), the domain is all real numbers except for values of x that make the denominator equal to zero since division by zero is undefined. To find these values, we need to solve the equation formed by setting the denominator equal to zero: x² + 6x + 8 = 0.
To solve for the values where the denominator is zero, we can factor the quadratic or use the quadratic formula. Factoring the quadratic, we get:
(x + 4)(x + 2) = 0,
which gives us the roots x = -4 and x = -2. These are the values that cannot be in the domain of the function. Hence, the domain of f(x) is all real numbers except x = -4 and x = -2.