To determine the values of [tex]\( x \)[/tex] for which the rational expression
[tex]\[
\frac{x+8}{x^2 - 2x - 24}
\][/tex]
is undefined, we need to focus on the denominator. A rational expression is undefined whenever its denominator is equal to zero because division by zero is undefined.
Let's find the values of [tex]\( x \)[/tex] that make the denominator zero. Consider the denominator [tex]\( x^2 - 2x - 24 \)[/tex]. We set it equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x^2 - 2x - 24 = 0
\][/tex]
To solve this quadratic equation, we can factor it. We need to find two numbers that multiply to [tex]\(-24\)[/tex] (the constant term) and add to [tex]\(-2\)[/tex] (the coefficient of the middle term). These two numbers are [tex]\(-6\)[/tex] and [tex]\(4\)[/tex]. So, we factor the quadratic expression:
[tex]\[
x^2 - 2x - 24 = (x - 6)(x + 4)
\][/tex]
Thus, we rewrite the equation as:
[tex]\[
(x - 6)(x + 4) = 0
\][/tex]
To find the values of [tex]\( x \)[/tex] that satisfy this equation, we set each factor equal to zero:
[tex]\[
x - 6 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
Solving these simple linear equations, we get:
[tex]\[
x = 6 \quad \text{or} \quad x = -4
\][/tex]
Hence, the rational expression is undefined for [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
Therefore, the correct answers are:
A. -4
B. 6
