1. **State the problem:** Simplify the expression $10x^2 \sqrt{x^5 - 3}$. 2. **Recall the formula and rules:** The square root of a product can be written as the product of square roots: $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ 3. **Rewrite the expression:** $$10x^2 \sqrt{x^5 - 3}$$ 4. **Note:** Since $x^5 - 3$ is inside the square root and cannot be simplified further without factoring or additional information, the expression is already simplified in terms of radicals. 5. **If we want to express $x^2$ inside the square root:** $$10x^2 \sqrt{x^5 - 3} = 10 \sqrt{x^4} \sqrt{x^5 - 3} = 10 \sqrt{x^4 (x^5 - 3)}$$ 6. **Combine under one square root:** $$10 \sqrt{x^4 (x^5 - 3)} = 10 \sqrt{x^{9} - 3x^4}$$ 7. **Final simplified form:** $$10 \sqrt{x^{9} - 3x^4}$$ This is a valid simplified form combining the terms under one radical. **Answer:** $$10 \sqrt{x^{9} - 3x^4}$$