To solve the problem, we need to utilize the given information that [tex]\((x + 5)\)[/tex] is a factor of the polynomial [tex]\( p(x) = x^3 - 20x + 5k \)[/tex]. This means that when [tex]\( x = -5 \)[/tex] is substituted into the polynomial, the result should be zero.
Here's the detailed step-by-step solution:
1. Substitute [tex]\( x = -5 \)[/tex] into [tex]\( p(x) \)[/tex]:
[tex]\[
p(-5) = (-5)^3 - 20(-5) + 5k
\][/tex]
2. Simplify the expression:
[tex]\[
(-5)^3 = -125
\][/tex]
[tex]\[
-20(-5) = 100
\][/tex]
Therefore,
[tex]\[
p(-5) = -125 + 100 + 5k
\][/tex]
3. Set the polynomial equal to zero because [tex]\((x + 5)\)[/tex] is a factor:
[tex]\[
-125 + 100 + 5k = 0
\][/tex]
4. Combine like terms:
[tex]\[
-125 + 100 = -25
\][/tex]
Therefore,
[tex]\[
-25 + 5k = 0
\][/tex]
5. Solve for [tex]\( k \)[/tex]:
[tex]\[
-25 + 5k = 0
\][/tex]
[tex]\[
5k = 25
\][/tex]
[tex]\[
k = \frac{25}{5}
\][/tex]
[tex]\[
k = 5
\][/tex]
So, the correct value of [tex]\( k \)[/tex] is [tex]\( 5 \)[/tex]. Hence, the answer is:
[tex]\[
\boxed{5}
\][/tex]
