To determine the values of [tex]\( x \)[/tex] for which the given rational expression
[tex]\[
\frac{x+8}{x^2 - 2x - 24}
\][/tex]
is undefined, we need to examine the denominator, [tex]\( x^2 - 2x - 24 \)[/tex]. The rational expression is undefined when the denominator is equal to zero, because division by zero is undefined.
Step-by-step, we solve for [tex]\( x \)[/tex] in the equation:
[tex]\[
x^2 - 2x - 24 = 0
\][/tex]
This is a quadratic equation, which we can solve by factoring. To factor [tex]\( x^2 - 2x - 24 \)[/tex], we look for two numbers that multiply to [tex]\(-24\)[/tex] and add to [tex]\(-2\)[/tex]. The numbers that fit these criteria are [tex]\( -6 \)[/tex] and [tex]\( 4 \)[/tex].
Thus, we can factor the quadratic as follows:
[tex]\[
x^2 - 2x - 24 = (x - 6)(x + 4) = 0
\][/tex]
Next, we set each factor equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[
x - 6 = 0 \quad \text{or} \quad x + 4 = 0
\][/tex]
[tex]\[
x = 6 \quad \text{or} \quad x = -4
\][/tex]
Therefore, the rational expression [tex]\(\frac{x+8}{x^2 - 2x - 24}\)[/tex] is undefined for [tex]\( x = 6 \)[/tex] and [tex]\( x = -4 \)[/tex].
So, the values of [tex]\( x \)[/tex] that make the expression undefined are:
[tex]\[
\boxed{6 \text{ and } -4}
\][/tex]
From the provided options, we should select options:
A. 6
C. -4
