1. **State the problem:** Simplify the expression $$(4r^3 + 3r^4) - (r^4 - 5r^3).$$ 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to each term inside the second parentheses: $$4r^3 + 3r^4 - r^4 + 5r^3.$$ 3. **Combine like terms:** Group terms with the same power of $r$: $$ (4r^3 + 5r^3) + (3r^4 - r^4).$$ 4. **Add coefficients:** $$ 9r^3 + 2r^4.$$ 5. **Write the final simplified expression:** $$2r^4 + 9r^3.$$