1. **State the problem:** Simplify the expression $$(a - b - 3c)(a + b - 3c) + (a - b)^2 + (2 - 3c)^2 + (6a + 2b)(a + c).$$ 2. **Use the distributive property and FOIL method:** - Expand each product separately. 3. **Expand $(a - b - 3c)(a + b - 3c)$:** $$= a(a + b - 3c) - b(a + b - 3c) - 3c(a + b - 3c)$$ $$= a^2 + ab - 3ac - ab - b^2 + 3bc - 3ac - 3bc + 9c^2$$ Simplify terms: $$= a^2 - b^2 - 6ac + 9c^2$$ 4. **Expand $(a - b)^2$:** $$= a^2 - 2ab + b^2$$ 5. **Expand $(2 - 3c)^2$:** $$= 4 - 12c + 9c^2$$ 6. **Expand $(6a + 2b)(a + c)$:** $$= 6a \cdot a + 6a \cdot c + 2b \cdot a + 2b \cdot c$$ $$= 6a^2 + 6ac + 2ab + 2bc$$ 7. **Sum all expanded parts:** $$ (a^2 - b^2 - 6ac + 9c^2) + (a^2 - 2ab + b^2) + (4 - 12c + 9c^2) + (6a^2 + 6ac + 2ab + 2bc) $$ 8. **Combine like terms:** - $a^2$ terms: $a^2 + a^2 + 6a^2 = 8a^2$ - $b^2$ terms: $-b^2 + b^2 = 0$ - $ab$ terms: $-2ab + 2ab = 0$ - $ac$ terms: $-6ac + 6ac = 0$ - $bc$ terms: $2bc$ - $c^2$ terms: $9c^2 + 9c^2 = 18c^2$ - Constants and $c$ terms: $4 - 12c$ 9. **Final simplified expression:** $$8a^2 + 2bc + 18c^2 + 4 - 12c$$ **Answer:** $$\boxed{8a^2 + 2bc + 18c^2 + 4 - 12c}$$