To solve the equation \((x+2)^4 + 2(y+3)^{10} = 0\), we can analyze its components.

1. The term \((x+2)^4\) is a fourth power, meaning it is always non-negative (i.e., \((x+2)^4 \geq 0\)).

2. The term \(2(y+3)^{10}\) is also non-negative since it is a non-negative constant multiplied by a tenth power (i.e., \(2(y+3)^{10} \geq 0\)).

Given both terms are non-negative, the only way for their sum to equal zero is if both terms are exactly zero:

\[
(x+2)^4 = 0 \quad \text{and} \quad 2(y+3)^{10} = 0
\]

### Solving for \(x\):
Starting with \((x+2)^4 = 0\):
\[
x+2 = 0 \implies x = -2
\]

### Solving for \(y\):
Now for \(2(y+3)^{10} = 0\):
\[
(y+3)^{10} = 0 \implies y+3 = 0 \implies y = -3
\]

### Conclusion:
The only solution to the equation \((x+2)^4 + 2(y+3)^{10} = 0\) is:
\[
\boxed{(-2, -3)}
\]