jmc

Instead of converting to base 10 and then to base 4, we use the fact that $2^2=4$. We have $11011000_2=1\cdot 2^7+1\cdot 2^6+1\cdot 2^4+1\cdot 2^3$ $=2\cdot (2^2{)}^3+1\cdot (2^2{)}^3+1\cdot (2^2{)}^2+2\cdot (2^2{)}^1$ $=3\cdot 4^3+1\cdot 4^2+2\cdot 4^1+0\cdot 4^0=\boxed{3120_4}$.