1. **State the problem:** Solve for the two possible values of $p$ in the equation $$\frac{78 - 3p^2}{5} = 6.$$\n\n2. **Use the formula and rules:** To solve for $p$, first eliminate the denominator by multiplying both sides by 5. Then isolate $p^2$ and solve the quadratic equation.\n\n3. **Step-by-step solution:**\nMultiply both sides by 5:\n$$78 - 3p^2 = 6 \times 5 = 30.$$\n\nSubtract 78 from both sides:\n$$-3p^2 = 30 - 78 = -48.$$\n\nDivide both sides by $-3$ to isolate $p^2$:\n$$p^2 = \frac{-48}{-3} = 16.$$\n\n4. **Find $p$ by taking the square root:**\n$$p = \pm \sqrt{16} = \pm 4.$$\n\n5. **Final answer:** The two possible values of $p$ are $4$ and $-4$.