jmc

Since $182$ is divisible by 14, the first term does not affect the residue of the whole expression, modulo 14. Since $15\cdot 7$ is divisible by 7 but not by 14 (since it isn't even), it has a residue of 7. So the residue of the sum is $$182\cdot 12-15\cdot 7+3\equiv 0-7+3\equiv -4\equiv \boxed{10}(\mathrm{mod}14).$$