1. **State the problem:** Simplify the expression $$(4x^2 + 1) - (2x^2 - 6x + 3)$$. 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to each term inside the second parentheses. $$4x^2 + 1 - 2x^2 + 6x - 3$$ 3. **Combine like terms:** Group terms with the same powers of $x$. $$ (4x^2 - 2x^2) + 6x + (1 - 3) $$ 4. **Simplify each group:** $$ 2x^2 + 6x - 2 $$ 5. **Final simplified expression:** $$2x^2 + 6x - 2$$