Step 1 :The Rational Root Theorem states that if a polynomial has a rational root, p/q (where p is a factor of the constant term and q is a factor of the leading coefficient), then p is a factor of the constant term and q is a factor of the leading coefficient.
Step 2 :The constant term in the given polynomial is -5 and the leading coefficient is 2.
Step 3 :The factors of -5 are ±1, ±5 and the factors of 2 are ±1, ±2.
Step 4 :So, the possible rational roots of the polynomial are ±1, ±5, ±1/2, ±5/2.
Step 5 :We can use synthetic division or direct substitution to test these possible roots.
Step 6 :By direct substitution, we find that -1, -1/2, and 5 are the roots of the polynomial.
Step 7 :\(\boxed{-1, -1/2, 5}\) are the zeros of the function \(h(x) = 2x^3 - 3x^2 - 12x - 5\).