Let's solve the given equation step by step. The equation is:

$4x - 5 - x^2 = q - (x + p)^2$

We need to find the values of $p$ and $q$, where they are integers.

Step 1: Expand the right-hand side

Expand $(x + p)^2$ on the right-hand side:

$(x + p)^2 = x^2 + 2px + p^2$

Substitute this into the equation:

$4x - 5 - x^2 = q - (x^2 + 2px + p^2)$

Step 2: Simplify the equation

Distribute the negative sign on the right-hand side:

$4x - 5 - x^2 = q - x^2 - 2px - p^2$

Now move all the terms involving $x$ and constants to one side of the equation:

$4x - 5 - x^2 + x^2 + 2px + p^2 = q$

This simplifies to:

$4x + 2px + (p^2 - 5) = q$

Step 3: Compare coefficients

Now compare the coefficients of like terms on both sides of the equation. On the left-hand side, the coefficient of $x$ is $4 + 2p$, and the constant term is $p^2 - 5$.

On the right-hand side, the equation is equal to $q$. Hence, we have two conditions:

1. $4 + 2p = 0$ (the coefficient of $x$ must be zero)
2. $p^2 - 5 = q$ (the constant term must equal $q$)

Step 4: Solve for $p$ and $q$

From $4 + 2p = 0$:

Solve for $p$:

$2p = -4 \quad \Rightarrow \quad p = -2$

Substitute $p = -2$ into $p^2 - 5 = q$:

$(-2)^2 - 5 = q \quad \Rightarrow \quad 4 - 5 = q \quad \Rightarrow \quad q = -1$

Final Answer:

The values of $p$ and $q$ are: $p = -2 \quad \text{and} \quad q = -1$

Let me know if you'd like more details, or if you have any further questions!

Here are some related questions to explore:

1. What happens if $p$ is a fraction instead of an integer?
2. How would you solve this if the powers of $x$ were higher than 2?
3. What role does comparing coefficients play in solving equations like this?
4. Can you explain why the method of comparing coefficients works?
5. How would the solution change if we added a term involving $x^3$ on both sides?

Tip: Always check your final answers by plugging them back into the original equation to ensure accuracy.