A term like [tex] x-k [/tex] is a factor of the polynomial [tex] p(x) [/tex] if and only if [tex] p(k)=0 [/tex]. Let's test this condition for all the given options: given [tex] p(x) = 4x^3 + 11x^2 − 75x + 18 [/tex], we have that
[tex] x-3 [/tex] is a factor of [tex] p(x) [/tex], because [tex] p(3) = 0 [/tex]
[tex] x+3 [/tex] is not a factor of [tex] p(x) [/tex], because [tex] p(-3) = 234 [/tex]
[tex] x-\frac{1}{3} [/tex] is not a factor of [tex] p(x) [/tex], because [tex] p\(left \frac{1}{3}) = -\frac{152}{27} [/tex]
[tex] x+\frac{1}{3} [/tex] is not a factor of [tex] p(x) [/tex], because [tex] p\(left -\frac{1}{3}) = \frac{1190}{27} [/tex]
So, the only correct answer is (x-3)
