Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.
**
   - Check the sum of coefficients:
     \[
     a_3 + a_2 + a_1 + a_0 = 2 + 0 + 1 + 4 = 7
     \]
   - Check the polynomial value at \( x = 10 \):
     \[
     p(10) = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10 + 4 = 2000 + 0 + 10 + 4 = 2014
     \]

Thus, the polynomial \( p(x) = 2x^3 + x + 4 \) satisfies all the given conditions.

The final answer is \( \boxed{ 2x^3 + x + 4 } \)