1. **State the problem:** We need to find expressions for $A$, $B$, and $C$ in the addition of algebraic expressions involving $d$ and $f$. Given the expressions: $$5d + 2f + 3d + 4f + A$$ $$B + 5d + 9f = C$$ We want to find what $A$, $B$, and $C$ must be so the addition is consistent, similar to the example with $a$. 2. **Identify $B$:** From the diagram, $B$ is the sum of $5d + 2f$ and $3d + 4f$. Adding like terms: $$B = (5d + 3d) + (2f + 4f) = 8d + 6f$$ 3. **Find $A$:** The expression $5d + 2f + 3d + 4f + A$ means adding $A$ to the sum $5d + 2f + 3d + 4f$. Since $B$ is defined as this sum without $A$, and the diagram shows $B$ plus $5d + 9f$ equals $C$, the unknown $A$ is the expression added to $3d + 4f$ to get $B$: $$A = 5d + 2f$$ This matches the left-hand addition sequence. 4. **Calculate $C$:** Given: $$B + 5d + 9f = C$$ Substitute $B$: $$C = (8d + 6f) + 5d + 9f = (8d + 5d) + (6f + 9f) = 13d + 15f$$ **Final answers:** $$A = 5d + 2f$$ $$B = 8d + 6f$$ $$C = 13d + 15f$$