1. **Problem statement:** Solve the system of equations: $$\frac{x + y + 3}{x - 2y} + \frac{x - 2y}{x + y + 3} = 2$$ $$x + 2y = 4$$ 2. **Formula and rules:** To solve this system, we will use substitution and algebraic manipulation. Note that the first equation involves fractions, so we will multiply through to clear denominators carefully. 3. **Step 1: Express $x$ from the second equation:** $$x = 4 - 2y$$ 4. **Step 2: Substitute $x$ into the first equation:** $$\frac{(4 - 2y) + y + 3}{(4 - 2y) - 2y} + \frac{(4 - 2y) - 2y}{(4 - 2y) + y + 3} = 2$$ Simplify numerator and denominator: $$\frac{7 - y}{4 - 4y} + \frac{4 - 4y}{7 - y} = 2$$ 5. **Step 3: Multiply both sides by the common denominator $(4 - 4y)(7 - y)$ to clear fractions:** $$ (7 - y)^2 + (4 - 4y)^2 = 2(4 - 4y)(7 - y) $$ 6. **Step 4: Expand each term:** $$ (7 - y)^2 = 49 - 14y + y^2 $$ $$ (4 - 4y)^2 = 16 - 32y + 16y^2 $$ $$ 2(4 - 4y)(7 - y) = 2(28 - 4y - 28y + 4y^2) = 2(28 - 32y + 4y^2) = 56 - 64y + 8y^2 $$ 7. **Step 5: Write the equation:** $$ 49 - 14y + y^2 + 16 - 32y + 16y^2 = 56 - 64y + 8y^2 $$ 8. **Step 6: Combine like terms on the left:** $$ 65 - 46y + 17y^2 = 56 - 64y + 8y^2 $$ 9. **Step 7: Bring all terms to one side:** $$ 65 - 46y + 17y^2 - 56 + 64y - 8y^2 = 0 $$ $$ (65 - 56) + (-46y + 64y) + (17y^2 - 8y^2) = 0 $$ $$ 9 + 18y + 9y^2 = 0 $$ 10. **Step 8: Divide entire equation by 9:** $$ \cancel{9} + 2y \times \cancel{9} + y^2 \times \cancel{9} = 0 \Rightarrow 1 + 2y + y^2 = 0 $$ 11. **Step 9: Rewrite as:** $$ y^2 + 2y + 1 = 0 $$ 12. **Step 10: Recognize perfect square:** $$ (y + 1)^2 = 0 $$ 13. **Step 11: Solve for $y$:** $$ y = -1 $$ 14. **Step 12: Substitute $y$ back into $x = 4 - 2y$:** $$ x = 4 - 2(-1) = 4 + 2 = 6 $$ 15. **Final answer:** $$ \boxed{(x, y) = (6, -1)} $$