1. **State the problem:** Simplify the expression $$\frac{2 + (x + iy) + (x + iy)^2}{(x + iy - 2)(x + iy - 3)(x + iy - 4)(x + iy - 5)}$$ where $x$ and $y$ are real numbers and $i$ is the imaginary unit. 2. **Recall the formula and rules:** Here, $z = x + iy$ is a complex number. We will simplify numerator and denominator separately. 3. **Simplify numerator:** $$2 + z + z^2$$ 4. **Simplify denominator:** $$(z - 2)(z - 3)(z - 4)(z - 5)$$ 5. **Expand denominator in pairs:** $$(z - 2)(z - 3) = z^2 - 5z + 6$$ $$(z - 4)(z - 5) = z^2 - 9z + 20$$ 6. **Multiply these two quadratics:** $$ (z^2 - 5z + 6)(z^2 - 9z + 20) $$ 7. **Expand:** $$ z^2 \cdot z^2 = z^4 $$ $$ z^2 \cdot (-9z) = -9z^3 $$ $$ z^2 \cdot 20 = 20z^2 $$ $$ -5z \cdot z^2 = -5z^3 $$ $$ -5z \cdot (-9z) = 45z^2 $$ $$ -5z \cdot 20 = -100z $$ $$ 6 \cdot z^2 = 6z^2 $$ $$ 6 \cdot (-9z) = -54z $$ $$ 6 \cdot 20 = 120 $$ 8. **Combine like terms:** $$ z^4 + (-9z^3 - 5z^3) + (20z^2 + 45z^2 + 6z^2) + (-100z - 54z) + 120 $$ $$ = z^4 - 14z^3 + 71z^2 - 154z + 120 $$ 9. **Final simplified expression:** $$ \frac{2 + z + z^2}{z^4 - 14z^3 + 71z^2 - 154z + 120} $$ 10. **Interpretation:** This is the simplified form of the original expression in terms of $z = x + iy$. Further factorization depends on specific values of $x$ and $y$ or context. **Answer:** $$\frac{2 + (x + iy) + (x + iy)^2}{(x + iy)^4 - 14(x + iy)^3 + 71(x + iy)^2 - 154(x + iy) + 120}$$