Answer
$-(x+5y)\sqrt[3]{2xy}$
Work Step by Step
Simplify each radical by factoring the radicand such that one of the factors is a perfect cube:
$\sqrt[3]{16x^4y}-3x\sqrt[3]{2xy}+5\sqrt[3]{-2xy^4}\\
=\sqrt[3]{8x^3\cdot 2xy}-3x\sqrt[3]{2xy}+5\sqrt[3]{-y^3\cdot 2xy}\\
=\sqrt[3]{(2x)^3\cdot 2xy}-3x\sqrt[3]{2xy}+5\sqrt[3]{(-y)^3\cdot 2xy}\\
$
Use the rule $\sqrt[n]{a\cdot b}=\sqrt[n]{a} \cdot \sqrt[n]{b}$ then simplify:
$=\sqrt[3]{(2x)^3}\cdot \sqrt[3]{2xy}-3x\sqrt[3]{2xy}+5\cdot\sqrt[3]{(-y)^3}\cdot \sqrt[3]{2xy}$
$=2x\cdot \sqrt[3]{2xy}-3x\sqrt[3]{2xy}+5(-y)\cdot \sqrt[3]{2xy}$
$=2x\sqrt[3]{2xy}-3x\sqrt[3]{2xy}-5y\sqrt[3]{2xy}$
Factor out $ \sqrt[3]{2xy}$ then simplify.
$=\sqrt[3]{2xy}(2x-3x-5y)$
$=\sqrt[3]{2xy}(-x-5y)$
$=(-x-5y)\sqrt[3]{2xy}$
$=(-1)(x+5y)\sqrt[3]{2xy}$
$=-(x+5y)\sqrt[3]{2xy}$
Hence, the correct answer is $-(x+5y)\sqrt[3]{2xy}$.
