**Finding the zeros of a polynomial**

To find the zeros of a polynomial, we need to solve the equation f(x) = 0, where f(x) is the given polynomial. In this case, the polynomial is 3x^4 + 2x^2 - 10x - 5.

**Given zeros**

We are given two zeros of the polynomial: root 5/3 and -root 5/3. Let's call these zeros a and b, respectively.

**Properties of zeros**

If a and b are zeros of a polynomial, then (x - a) and (x - b) are factors of the polynomial.

**Factoring the polynomial**

Since we know two zeros of the polynomial, we can write the polynomial as a product of its factors:

f(x) = (x - a)(x - b)(c1x^2 + c2x + c3)

Here, c1, c2, and c3 are constants that represent the remaining quadratic factor of the polynomial.

**Expanding the factors**

Expanding the factors, we get:

f(x) = (x - a)(x - b)(c1x^2 + c2x + c3)
= (x^2 - bx - ax + ab)(c1x^2 + c2x + c3)
= (x^2 - (a + b)x + ab)(c1x^2 + c2x + c3)
= c1x^4 + (c2 - a - b)c1x^3 + (c3 - c1(a + b) - ab)c1x^2 + (c3(a + b) - c2ab)x - abc3

**Comparing coefficients**

Since the polynomial is given as 3x^4 + 2x^2 - 10x - 5, we can compare the corresponding coefficients to find the values of c1, c2, and c3:

c1 = 3
c2 - a - b = 0
c3 - c1(a + b) - ab = 2
c3(a + b) - c2ab = -10
-abc3 = -5

**Substituting the given zeros**

Substituting the values of a = root 5/3 and b = -root 5/3, we get:

c2 - root 5/3 + root 5/3 = 0
c3 - 3(root 5/3)(-root 5/3) = 2
3c3(root 5/3) + 10(root 5/3)(-root 5/3) = -10

Simplifying these equations, we find:

c2 = 0
c3 = 2
c3 = -10

**Finding the remaining zeros**

Now that we have the quadratic factor of the polynomial, we can find the remaining zeros by solving the equation c1x^2 + c2x + c3 = 0:

3x^2 + 0x + 2 = 0

Using the quadratic formula, we find:

x = (-b ± √(b^2 - 4ac)) / (2a)
= (± √(-24)) / 6
= ± √(6)i