Introduction:
In this problem, we are given a polynomial and we need to find its zeros. Zeros of a polynomial are the values of the variable for which the polynomial becomes zero. Additionally, we need to verify the relationship between the zeros and the coefficients of the polynomial.

Given Polynomial:
The given polynomial is 4x^2 - 9.

Finding Zeros:
To find the zeros of the polynomial, we need to set the polynomial equal to zero and solve for x. So, we have:

4x^2 - 9 = 0

Factoring:
To factor the quadratic equation, we need to express it as a product of two binomials. In this case, we have a difference of squares.

4x^2 - 9 = (2x)^2 - 3^2

Using the difference of squares formula, we can rewrite the equation as:

(2x - 3)(2x + 3) = 0

Zero Product Property:
According to the zero product property, if a product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x.

2x - 3 = 0 or 2x + 3 = 0

Solving the first equation:
2x = 3
x = 3/2

Solving the second equation:
2x = -3
x = -3/2

Zeros of the Polynomial:
The zeros of the polynomial 4x^2 - 9 are x = 3/2 and x = -3/2.

Relationship between Zeros and Coefficients:
In a quadratic polynomial of the form ax^2 + bx + c, the sum of the zeros is given by -b/a and the product of the zeros is given by c/a.

For the given polynomial 4x^2 - 9, the sum of the zeros is:
-(-b/a) = 0/4 = 0

The product of the zeros is:
c/a = -9/4

Hence, the relationship between the zeros and the coefficients of the polynomial is as follows:
- The sum of the zeros is 0.
- The product of the zeros is -9/4.

Summary:
The zeros of the polynomial 4x^2 - 9 are x = 3/2 and x = -3/2. The relationship between the zeros and the coefficients of the polynomial is that the sum of the zeros is 0 and the product of the zeros is -9/4.