1. The problem asks to find the value of $q^2 - 2pq$ where $p$ is the minimum value and $q$ is the maximum value of the function $$f(x) = 3 \sin\left(x - \frac{\pi}{4}\right) + 5.$$\n\n2. The sine function $\sin(\theta)$ has a range of $[-1,1]$.\n\n3. Therefore, the minimum value of $f(x)$ occurs when $\sin\left(x - \frac{\pi}{4}\right) = -1$, and the maximum value occurs when $\sin\left(x - \frac{\pi}{4}\right) = 1$.\n\n4. Calculate the minimum value $p$:\n$$p = 3 \times (-1) + 5 = -3 + 5 = 2.$$\n\n5. Calculate the maximum value $q$: \n$$q = 3 \times 1 + 5 = 3 + 5 = 8.$$\n\n6. Now compute $q^2 - 2pq$: \n$$q^2 - 2pq = 8^2 - 2 \times 2 \times 8 = 64 - 32 = 32.$$\n\n7. The value that satisfies the expression is 32, which corresponds to option A.