To determine the value of [tex]\( k \)[/tex] such that [tex]\( (x+2) \)[/tex] is a factor of the polynomial [tex]\( x^3 - 6x^2 + kx + 10 \)[/tex], we can use the factor theorem. According to the factor theorem, if [tex]\( (x+2) \)[/tex] is a factor of the polynomial, then the polynomial should evaluate to zero when [tex]\( x = -2 \)[/tex].
Let's denote the polynomial by [tex]\( f(x) = x^3 - 6x^2 + kx + 10 \)[/tex].
By the factor theorem:
[tex]\[ f(-2) = 0 \][/tex]
Substitute [tex]\( x = -2 \)[/tex] into the polynomial:
[tex]\[ (-2)^3 - 6(-2)^2 + k(-2) + 10 = 0 \][/tex]
Compute each term:
[tex]\[ -8 - 6(4) - 2k + 10 = 0 \][/tex]
Simplify:
[tex]\[ -8 - 24 - 2k + 10 = 0 \][/tex]
[tex]\[ -22 - 2k = 0 \][/tex]
Isolate [tex]\( k \)[/tex]:
[tex]\[ -2k = 22 \][/tex]
[tex]\[ k = -11 \][/tex]
Therefore, the correct value for [tex]\( k \)[/tex] is [tex]\( \boxed{-11} \)[/tex].
