1. **State the problem:** Multiply out and simplify the expression $$(4x - 10\sqrt{x})(2x + 5\sqrt{x} - 7).$$ 2. **Recall the distributive property:** To multiply two expressions, multiply each term in the first expression by each term in the second expression. 3. **Multiply each term:** - $4x \times 2x = 8x^2$ - $4x \times 5\sqrt{x} = 20x\sqrt{x}$ - $4x \times (-7) = -28x$ - $-10\sqrt{x} \times 2x = -20x\sqrt{x}$ - $-10\sqrt{x} \times 5\sqrt{x} = -50x$ - $-10\sqrt{x} \times (-7) = 70\sqrt{x}$ 4. **Write the expanded expression:** $$8x^2 + 20x\sqrt{x} - 28x - 20x\sqrt{x} - 50x + 70\sqrt{x}$$ 5. **Combine like terms:** - Combine $20x\sqrt{x}$ and $-20x\sqrt{x}$: $$20x\sqrt{x} + (-20x\sqrt{x}) = \cancel{20x\sqrt{x}} + \cancel{-20x\sqrt{x}} = 0$$ - Combine $-28x$ and $-50x$: $$-28x - 50x = -78x$$ 6. **Simplified expression:** $$8x^2 - 78x + 70\sqrt{x}$$ **Final answer:** $$\boxed{8x^2 - 78x + 70\sqrt{x}}$$