1. **State the problem:** Simplify the expression $$a^2 (2a + 5a^3 + 1) - 10a$$. 2. **Apply the distributive property:** Multiply $$a^2$$ by each term inside the parentheses: $$a^2 \times 2a = 2a^{3}$$ $$a^2 \times 5a^3 = 5a^{5}$$ $$a^2 \times 1 = a^{2}$$ So the expression becomes: $$2a^{3} + 5a^{5} + a^{2} - 10a$$ 3. **Rearrange terms in descending powers of $$a$$:** $$5a^{5} + 2a^{3} + a^{2} - 10a$$ 4. **Final simplified expression:** $$\boxed{5a^{5} + 2a^{3} + a^{2} - 10a}$$ This is the simplified form; no like terms can be combined further because all powers of $$a$$ are distinct.