1. **State the problem:** Simplify the expression $5x \times \left(\frac{2}{5}x + 5\right)$.\n\n2. **Use the distributive property:** Multiply $5x$ by each term inside the parentheses:\n$$5x \times \frac{2}{5}x + 5x \times 5$$\n\n3. **Simplify each term:**\n- For the first term: $$5x \times \frac{2}{5}x = \cancel{5}x \times \frac{2}{\cancel{5}}x = 2x^2$$\n- For the second term: $$5x \times 5 = 25x$$\n\n4. **Write the simplified expression:**\n$$2x^2 + 25x$$\n\n5. **Final answer:** The simplified form of $5x \times \left(\frac{2}{5}x + 5\right)$ is $$\boxed{2x^2 + 25x}$$.