We are asked to find the linear factors of the polynomial:
f(x) = x^3 + 4 x^2 + 25 x + 100
So we start by trying to find a real zero for it using the rational zeros theorem.
We find that the rational number "-4) is one root of the polynomial, since when replacing it for "x" we get that:
f(-4) = 0
So we know that (x- -4) =(x + 4) is a binomial factor of the original polynomial.
Now, by using long division, we find the quadratic polynomial quotient:
f(x) = (x + 4) * (x^2 + 25)
So, now this quadratic factor (x^2 + 25) doesn't have real roots, but COMPLEX roots:
x^2 + 25 = 0
solving for x:
x^2 = - 25
Then the possible solutions are:
x = 5 i and x = -5i
which gives us the linear factors: ( x - 5 i ) and (x + 5 i)
Then the total factorization in linear terms is:
f(x) = (x + 4) (x - 5i) (x+ 5 i)
which agrees with option "C" in the provided list of possible solutions.