1. **State the problem:** Simplify and expand the expression $\left(2\sqrt{x} + \frac{1}{\sqrt{x}}\right) x^5$ using algebraic manipulation and the binomial theorem if applicable. 2. **Rewrite the expression:** Note that $\sqrt{x} = x^{\frac{1}{2}}$ and $\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}$. So the expression becomes: $$\left(2x^{\frac{1}{2}} + x^{-\frac{1}{2}}\right) x^5$$ 3. **Distribute $x^5$ to each term:** $$2x^{\frac{1}{2}} \cdot x^5 + x^{-\frac{1}{2}} \cdot x^5 = 2x^{5 + \frac{1}{2}} + x^{5 - \frac{1}{2}}$$ 4. **Simplify the exponents:** $$2x^{\frac{11}{2}} + x^{\frac{9}{2}}$$ 5. **Final simplified expression:** $$2x^{\frac{11}{2}} + x^{\frac{9}{2}}$$ **Note:** The binomial theorem is typically used for expanding powers of binomials like $(a+b)^n$. Here, the expression is a product, not a power, so direct distribution and exponent rules are used instead.