since complex roots only occur in complex conjugate pairs if 5i is root that - 5i is root as well

f(x) = C(x - 4)(x - 5i)(x + 5i), where C is constant that we need to find

f(x) = C(x - 4)(x^2 + 25)

f(x) = C(x^3 - 4x^2 + 25x - 100)

f(2) = 116

f(2) = C(2^3 - 4*2^2 + 25*2 - 100)

C(2^3 - 4*2^2 + 25*2 - 100) = 116

by solving we find:

C = -2

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the 3rd-degree polynomial function with real coefficients is:

f(x) = (-2)(x^3 - 4x^2 + 25x - 100)

f(x) = - 2x^3 + 8x^2 - 50x + 200