1. **State the problem:** Given functions $f(x) = 5x^{4} + 10$ and $g(x) = \sqrt{x}$, find the expression for $\frac{f}{g}(x)$ for $x \neq 0$. 2. **Recall the formula:** The division of two functions is given by $$\frac{f}{g}(x) = \frac{f(x)}{g(x)}$$ 3. **Substitute the given functions:** $$\frac{f}{g}(x) = \frac{5x^{4} + 10}{\sqrt{x}}$$ 4. **Rewrite the denominator using exponents:** $$\sqrt{x} = x^{\frac{1}{2}}$$ 5. **Divide each term in the numerator by $x^{\frac{1}{2}}$ separately:** $$\frac{5x^{4}}{x^{\frac{1}{2}}} + \frac{10}{x^{\frac{1}{2}}}$$ 6. **Apply the exponent subtraction rule $\frac{a^{m}}{a^{n}} = a^{m-n}$:** $$5x^{4 - \frac{1}{2}} + 10x^{-\frac{1}{2}}$$ 7. **Simplify the exponents:** $$5x^{\frac{8}{2} - \frac{1}{2}} + 10x^{-\frac{1}{2}} = 5x^{\frac{7}{2}} + 10x^{-\frac{1}{2}}$$ 8. **Final answer:** $$\frac{f}{g}(x) = 5x^{\frac{7}{2}} + 10x^{-\frac{1}{2}}$$ This corresponds to option (a).