To solve the given equation, let's first rewrite it clearly:

\[
\frac{X + 5}{X - 3} + 2 = \frac{2}{X - 3}
\]

To combine the fractions on the left-hand side, we need a common denominator. The common denominator is \(X - 3\). So, we rewrite \(2\) as \(\frac{2(X - 3)}{X - 3}\):

\[
\frac{X + 5}{X - 3} + \frac{2(X - 3)}{X - 3} = \frac{2}{X - 3}
\]

Now, combine the fractions on the left-hand side:

\[
\frac{X + 5 + 2(X - 3)}{X - 3} = \frac{2}{X - 3}
\]

Simplify the numerator:

\[
X + 5 + 2X - 6 = 3X - 1
\]

So, the equation becomes:

\[
\frac{3X - 1}{X - 3} = \frac{2}{X - 3}
\]

Since the denominators are the same, we can equate the numerators:

\[
3X - 1 = 2
\]

Solve for \(X\):

\[
3X - 1 = 2 \\
3X = 3 \\
X = 1
\]

So, the solution to the equation is \(X = 1\).