Let's simplify the expression \( (10 - \sqrt{49} + 1)^2 \).
First, we calculate \( \sqrt{49} \):
\[ \sqrt{49} = 7 \]
Now substitute this back into the expression:
\[ 10 - \sqrt{49} + 1 = 10 - 7 + 1 \]
Now compute \( 10 - 7 + 1 \):
\[ 10 - 7 = 3 \] \[ 3 + 1 = 4 \]
Now we have:
\[ (10 - \sqrt{49} + 1)^2 = (4)^2 \]
Finally, calculate \( 4^2 \):
\[ 4^2 = 16 \]
Therefore, the simplified expression is:
\[ \boxed{16} \]