1. **State the problem:** Simplify and analyze the expression $abc + 2fgh - af2 - bg2 - cf2 = 0$. 2. **Identify terms and variables:** The expression contains variables $a,b,c,f,g,h$ and constants. Note that $af2$ means $a \times f \times 2$, similarly for $bg2$ and $cf2$. 3. **Rewrite the expression with explicit multiplication:** $$abc + 2fgh - 2af - 2bg - 2cf = 0$$ 4. **Group terms to factor if possible:** Group the terms involving $a,b,c$: $$abc - 2af - 2bg - 2cf + 2fgh = 0$$ 5. **Look for common factors:** - From $abc - 2af$, factor $a$: $$a(bc - 2f)$$ - From $-2bg - 2cf$, factor $-2$: $$-2(bg + cf)$$ 6. **Rewrite the expression:** $$a(bc - 2f) - 2(bg + cf) + 2fgh = 0$$ 7. **No further simple factorization is apparent without additional context or constraints.** **Final simplified form:** $$a(bc - 2f) - 2(bg + cf) + 2fgh = 0$$