The question appears to ask for the minimum value of the expression:

$p^2 - 16pq + 64q^2 - 12$

We can simplify this expression by recognizing the terms $p^2 - 16pq + 64q^2$. This part resembles a perfect square. Let's rewrite it as follows:

$p^2 - 16pq + 64q^2 = (p - 8q)^2$

Therefore, the expression becomes:

$(p - 8q)^2 - 12$

Since $(p - 8q)^2 \geq 0$ for all real values of $p$ and $q$, the minimum value of $(p - 8q)^2$ is 0. Substituting 0 for $(p - 8q)^2$ gives:

$(p - 8q)^2 - 12 = 0 - 12 = -12$

Thus, the minimum value of the expression is $-12$.

Answer:

The correct option is B) -12.

Would you like a more detailed explanation, or do you have any questions?

Here are five related questions for further practice:

1. How do you identify a perfect square in a quadratic expression?
2. What are the general steps to complete the square for a quadratic expression?
3. How can we find the minimum or maximum value of a quadratic expression?
4. What other methods can be used to simplify quadratic expressions?
5. How does the discriminant relate to finding the minimum or maximum value?

Tip: Recognizing perfect squares is a helpful skill for quickly simplifying and solving quadratic expressions.