1. Use the relativistic energy-momentum relation

For a particle of rest mass $m$, momentum $p$, and total energy $E$, the correct relation is

2. Check the dimensions of each term

• $E^2$ has dimensions of energy squared.
• $p^2c^2$ has dimensions: $[p]^2[c]^2 = \left(\frac{\text{kg m}}{\text{s}}\right)^2 \left(\frac{\text{m}}{\text{s}}\right)^2 = \text{kg}^2\text{m}^4\text{s}^{-4}$ which is indeed energy squared.
• $m^2c^4$ also has dimensions of energy squared.

So this formula is dimensionally correct.

3. Evaluate the options

• A: $E^2 = pc^2 + m^2c^4$
  Incorrect, because the first term should be $p^2c^2$, not $pc^2$.

• B: $E^2 = pc^2 + m^2c^2$
  Incorrect, both terms are dimensionally wrong.

• C: $E^2 = p^2c^2 + m^2c^4$
  Correct.

• D: $E^2 = p^2c^2 + m^2c^2$
  Incorrect, the mass term should be $m^2c^4$.

4. Final answer

The correct equation is

So, the correct option is C.