Express in simplest radical form. \[ -4 x \sqrt{48 x}+\sqrt{3 x^{3}} \]

To simplify the expression \(-4 x \sqrt{48 x}+\sqrt{3 x^{3}}\), let's tackle each term separately. First, simplify \(\sqrt{48 x}\): \[ \sqrt{48 x} = \sqrt{48} \cdot \sqrt{x} = \sqrt{16 \cdot 3} \cdot \sqrt{x} = 4 \sqrt{3} \sqrt{x} = 4 \sqrt{3x} \] Now, substitute back into the first term: \[ -4 x \sqrt{48 x} = -4 x (4 \sqrt{3x}) = -16 x \sqrt{3x} \] Next, simplify \(\sqrt{3 x^3}\): \[ \sqrt{3 x^3} = \sqrt{3} \cdot \sqrt{x^3} = \sqrt{3} \cdot x^{3/2} = x^{3/2} \sqrt{3} \] Now that we have both terms simplified, combine them: \[ -16 x \sqrt{3x} + x^{3/2} \sqrt{3} \] Rewriting \(x^{3/2}\) gives us \(x\sqrt{x}\): \[ \sqrt{3} x^{3/2} = \sqrt{3} x \sqrt{x} = \sqrt{3x}(x) \] Now, our expression looks like: \[ -16 x \sqrt{3x} + \sqrt{3} x \sqrt{x} = x \sqrt{3x}(-16 + 1) \] This simplifies to: \[ -15 x \sqrt{3x} \] Thus, the expression in simplest radical form is: \[ \boxed{-15 x \sqrt{3x}} \]