Step-by-step explanation:

Let's break down the expression step by step:

1. You have:

- \(X = 3\sqrt{28}\)

- \(Y = 3\sqrt{27}\)

2. First, simplify \(Y\):

- \(Y = 3\sqrt{27} = 3 \cdot \sqrt{3^3} = 3 \cdot 3 = 9\)

3. Now, we have the values of \(X\) and \(Y\):

- \(X = 3\sqrt{28}\)

- \(Y = 9\)

4. Calculate \(X^2\):

- \(X^2 = (3\sqrt{28})^2 = 3^2 \cdot (\sqrt{28})^2 = 9 \cdot 28 = 252\)

5. Calculate \(XY\):

- \(XY = (3\sqrt{28}) \cdot 9 = 27\sqrt{28}\)

6. Calculate \(Y^2\):

- \(Y^2 = 9^2 = 81\)

7. Now, we can substitute these values into the expression:

\(\frac{X + Y - 1}{X^2 + XY + Y^2} = \frac{(3\sqrt{28}) + 9 - 1}{252 + 27\sqrt{28} + 81}\)

8. Simplify the numerator:

\(3\sqrt{28} + 9 - 1 = 3\sqrt{28} + 8\)

9. Now, rewrite the expression with the simplified values:

\(\frac{3\sqrt{28} + 8}{252 + 27\sqrt{28} + 81}\)

10. Combine the constants in the denominator:

\(252 + 81 = 333\)

11. Rewrite the expression:

\(\frac{3\sqrt{28} + 8}{333 + 27\sqrt{28}}\)

This is the simplified expression, but it cannot be further simplified without using an approximate numerical value for \(\sqrt{28}\) since it's not a perfect square. If you'd like an approximate numerical value for this expression, you can calculate it using a calculator or software.