1. **State the problem:** Simplify the algebraic expression $4x2p - 9y - 40xP - 4y3q + 2 + 100xy6q + 3$. 2. **Identify terms and variables:** The expression contains terms with variables $x$, $p$, $y$, $P$, and $q$. Note that $p$ and $P$ are different variables due to case sensitivity. 3. **Rewrite terms clearly:** - $4x2p$ means $4 \times x \times 2 \times p = 8xp$ - $-9y$ stays as is - $-40xP$ stays as is - $-4y3q$ means $-4 \times y \times 3 \times q = -12yq$ - $+2$ constant - $+100xy6q$ means $100 \times x \times y \times 6 \times q = 600xyq$ - $+3$ constant 4. **Rewrite the expression:** $$8xp - 9y - 40xP - 12yq + 2 + 600xyq + 3$$ 5. **Combine like terms:** - Constants: $2 + 3 = 5$ - No other like terms with same variables and powers. 6. **Final simplified expression:** $$8xp - 9y - 40xP - 12yq + 600xyq + 5$$ This is the simplified form as no further combination is possible due to different variables or powers.