To solve the equation $F(x) + F(4) = x^2 - 16 + F(1)$, we can rearrange it to isolate $F(x)$:

$F(x) = x^2 - 16 + F(1) - F(4)$

This means $F(x)$ is expressed in terms of $x$ and constants $F(1)$ and $F(4)$. If you want to delve deeper into the function's behavior or find specific values, you need to provide more information about $F(1)$ and $F(4)$ or any conditions for $F(x)$.

Do you need further details or assistance with a specific part of this problem? Here are some related questions:

1. What are the values of $F(1)$ and $F(4)$?
2. Is there a specific form or property of $F(x)$ you need to consider?
3. Would you like to explore specific values of $x$?
4. Are there any constraints on $F(x)$?
5. Do you need help with another related equation or function?

Tip: Always consider substituting known values to simplify functions in equations.