Simplify each radical:

$4\sqrt{24} = 4\sqrt{4 \cdot 6} = 4 \cdot 2\sqrt{6} = 8\sqrt{6}$

$5\sqrt{54} = 5\sqrt{9 \cdot 6} = 5 \cdot 3\sqrt{6} = 15\sqrt{6}$

Subtract:

$8\sqrt{6} - 15\sqrt{6} = -7\sqrt{6}$