Carly made her first error when completing the square. Based on her steps:

1. Carly starts with $x(x + 4) = 117$.
2. Expanding the equation gives $x^2 + 4x = 117$.
3. She completes the square by adding $4^2 = 16$ to both sides: $x^2 + 4x + 4^2 = 117 + 4^2$.

This step is correct. The addition of $16$ is valid for completing the square.

However, the error appears in the next step, $(x + 4)^2 = 133$, when solving $x + 4 = \pm \sqrt{133}$. Carly writes the final solution as $x = -4 \pm \sqrt{133}$. The sign of $\pm$ is correct here, meaning option C is not the error.

The correct first error in her work is **option D: She should have subtracted the value from the right side of the equation instead of adding. The term likely meant misbalance numerical.