1. **State the problem:** Simplify the polynomial expression given by combining like terms: $$8mr^5 - 2rn^6 + 5mr^2n^4 - mr^3n^3 + n^6 - 4m^6 + 9rn^2n^4 - mn^5 - 4m^3rn^3$$ 2. **Identify like terms:** - Note that $9rn^2n^4 = 9rn^{6}$ since $n^2 \cdot n^4 = n^{6}$. - Group terms by their variables and powers: - Terms with $n^6$: $-2rn^6 + 9rn^6 + n^6$ - Terms with $mr^5$: $8mr^5$ - Terms with $mr^2n^4$: $5mr^2n^4$ - Terms with $mr^3n^3$: $-mr^3n^3 - 4m^3rn^3$ - Terms with $mn^5$: $-mn^5$ - Terms with $m^6$: $-4m^6$ 3. **Simplify like terms:** - Combine $-2rn^6 + 9rn^6 = 7rn^6$ - So the $n^6$ terms become $7rn^6 + n^6$ 4. **Rewrite the expression:** $$8mr^5 + 5mr^2n^4 - mr^3n^3 - 4m^3rn^3 + 7rn^6 + n^6 - mn^5 - 4m^6$$ 5. **Check for further simplification:** - The terms $-mr^3n^3$ and $-4m^3rn^3$ are not like terms because of different powers of $m$ and $r$. - No other like terms to combine. 6. **Final simplified expression:** $$8mr^5 + 5mr^2n^4 - mr^3n^3 - 4m^3rn^3 + 7rn^6 + n^6 - mn^5 - 4m^6$$