To solve the equation

\[
(2x - 5)^{2024} + (3y + 4)^{2016} = 0,
\]

we note that both terms in the equation are raised to even powers. Therefore, the expressions \((2x - 5)^{2024}\) and \((3y + 4)^{2016}\) are both non-negative. Specifically, they are zero or positive for all values of \(x\) and \(y\):

- For \((2x - 5)^{2024} \geq 0\)
- For \((3y + 4)^{2016} \geq 0\)

Since the sum of two non-negative terms equals zero, both terms must independently be equal to zero:

1. \( (2x - 5)^{2024} = 0 \)
2. \( (3y + 4)^{2016} = 0 \)

We can solve these equations separately.

### Step 1: Solve \( (2x - 5)^{2024} = 0 \)

The expression is zero when:

\[
2x - 5 = 0
\]

Solving for \(x\):

\[
2x = 5 \implies x = \frac{5}{2} = 2.5.
\]

### Step 2: Solve \( (3y + 4)^{2016} = 0 \)

The expression is zero when:

\[
3y + 4 = 0.
\]

Solving for \(y\):

\[
3y = -4 \implies y = -\frac{4}{3}.
\]

### Final Solution

Thus, the solution to the equation \( (2x - 5)^{2024} + (3y + 4)^{2016} = 0 \) is:

\[
x = \frac{5}{2}, \quad y = -\frac{4}{3}.
\]