Solution:

To find the value of k in the quadratic polynomial f(x) = 4x^2 - 8kx - 9, we need to determine when f(x) is the negative of another quadratic polynomial.

Step 1: Equating the two quadratic polynomials
Let's assume the other quadratic polynomial is g(x). Since f(x) is the negative of g(x), we can write it as -g(x). Therefore, we can equate the two polynomials as follows:

f(x) = -g(x)
4x^2 - 8kx - 9 = -g(x)

Step 2: Comparing the coefficients
In order for f(x) to be the negative of g(x), the corresponding coefficients of both polynomials should be opposite in sign. Let's compare the coefficients:

Comparing the coefficient of x^2:
4 = -1 (opposite signs)

Comparing the coefficient of x:
-8k = 0 (opposite signs)

Comparing the constant term:
-9 = 0 (not opposite signs)

Step 3: Solving for k
From the comparison of coefficients, we can see that the coefficient of the constant term does not have opposite signs. This means that there is no value of k that satisfies the condition of f(x) being the negative of another quadratic polynomial.

Therefore, there is no value of k that can make f(x) the negative of another quadratic polynomial.

Conclusion:
After analyzing the quadratic polynomial f(x) = 4x^2 - 8kx - 9, we found that there is no value of k that would make f(x) the negative of another quadratic polynomial.