To solve this, we need to find which expression, when added to $5 - p^2 - q^2$, will yield zero.

Step-by-step solution:

1. The given expression is: $5 - p^2 - q^2$

2. To find what should be added to this expression to make the result zero, we solve: $(5 - p^2 - q^2) + x = 0$ This simplifies to: $x = -(5 - p^2 - q^2) = -5 + p^2 + q^2$

3. Therefore, the correct expression that should be added to $5 - p^2 - q^2$ is: $-5 + p^2 + q^2$ This corresponds to option C.

So, the correct answer is C.

Would you like further details on any part of this solution?

Here are 5 related questions:

1. What happens if we add or subtract the same variable terms from both sides of an equation?
2. How do we simplify expressions with multiple variables?
3. Can you explain how to distribute negative signs across parentheses in algebraic expressions?
4. What are some common mistakes to avoid when solving equations involving squares?
5. How can we verify if our solution to an equation is correct?

Tip: Always double-check your sign changes when solving algebraic expressions, especially when distributing negatives.