To solve the equation

\[
\frac{X + 3}{X - 2} + \frac{X + 2}{X - 3} = 2,
\]

we need to find a common denominator for the fractions on the left-hand side. The common denominator is \((X - 2)(X - 3)\). Rewriting each fraction with this common denominator, we get:

\[
\frac{(X + 3)(X - 3)}{(X - 2)(X - 3)} + \frac{(X + 2)(X - 2)}{(X - 2)(X - 3)} = 2.
\]

Simplify the numerators:

\[
\frac{X^2 - 9}{(X - 2)(X - 3)} + \frac{X^2 - 4}{(X - 2)(X - 3)} = 2.
\]

Combine the fractions:

\[
\frac{X^2 - 9 + X^2 - 4}{(X - 2)(X - 3)} = 2.
\]

Simplify the numerator:

\[
\frac{2X^2 - 13}{(X - 2)(X - 3)} = 2.
\]

To clear the fraction, multiply both sides by \((X - 2)(X - 3)\):

\[
2X^2 - 13 = 2(X - 2)(X - 3).
\]

Expand the right-hand side:

\[
2X^2 - 13 = 2(X^2 - 5X + 6).
\]

Distribute the 2 on the right-hand side:

\[
2X^2 - 13 = 2X^2 - 10X + 12.
\]

Subtract \(2X^2\) from both sides:

\[
-13 = -10X + 12.
\]

Add 10X to both sides:

\[
10X - 13 = 12.
\]

Add 13 to both sides:

\[
10X = 25.
\]

Divide by 10:

\[
X = \frac{25}{10} = 2.5.
\]

So, the solution to the equation is:

\[
X = 2.5.
\]