Introduction:
In this problem, we are given a quadratic polynomial f(x) = x^2 - 3x - 2 and we need to find a quadratic polynomial whose zeros are 1/2alpha beta and 1/2beta alpha, where alpha and beta are the zeros of f(x).

Solution:
To find the zeros of f(x), we can use the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / 2a
where a = 1, b = -3, and c = -2.

Solving for x, we get:
alpha = (3 + sqrt(17)) / 2
beta = (3 - sqrt(17)) / 2

Step 1: Finding the product of the zeros:
Let's first find the product of the zeros of f(x):
alpha * beta = [(3 + sqrt(17)) / 2] * [(3 - sqrt(17)) / 2]
= (9 - 17) / 4
= -2

Step 2: Finding the sum of the zeros:
Next, let's find the sum of the zeros of f(x):
alpha + beta = (3 + sqrt(17)) / 2 + (3 - sqrt(17)) / 2
= 3

Step 3: Writing a quadratic polynomial with the given zeros:
We can write a quadratic polynomial with the given zeros using the following formula:
(x - p)(x - q) = 0
where p and q are the zeros of the new polynomial.

Let's find p and q:
p = 1/2alpha beta = -1/alpha
q = 1/2beta alpha = -1/beta

Substituting these values in the formula, we get:
(x + 1/alpha)(x + 1/beta) = 0

Multiplying both sides by alpha * beta, we get:
(alpha * beta * x + beta)(alpha * beta * x + alpha) = 0

Expanding and simplifying, we get:
x^2 + (alpha + beta) * x + alpha * beta = 0

Substituting the values of alpha + beta and alpha * beta, we get:
x^2 + 3x - 2 = 0

Therefore, the quadratic polynomial whose zeros are 1/2alpha beta and 1/2beta alpha is:
x^2 + 3x - 2 = 0.

Conclusion:
In this problem, we used the quadratic formula to find the zeros of the given polynomial, and then used the formula for finding a quadratic polynomial with given zeros to find the desired polynomial.