To simplify the expression

\[
M = \frac{a}{ab + a + 1} + \frac{b}{ba + b + 1} + \frac{2C}{ac + 2c + 2}
\]

given that \(abc = 2\), let's work on each term step by step.

1. **First term**:

\[
\frac{a}{ab + a + 1} = \frac{a}{a(b + 1) + 1}
\]

2. **Second term**:

\[
\frac{b}{ba + b + 1} = \frac{b}{b(a + 1) + 1}
\]

3. **Third term**:

\[
\frac{2C}{ac + 2c + 2} = \frac{2C}{2(c + 1) + ac} = \frac{C}{c + 1 + \frac{ac}{2}}
\]

Now, using \(abc = 2\) can help simplify these fractions further.

Instead of proceeding with algebraic manipulation at this step, let's denote

- \(x = a + 1\)
- \(y = b + 1\)
- \(z = c + 1\)

Using these substitutions helps in consolidating terms. We can then rewrite the denominators and potentially find common denominators for addition.

### To simplify:

Combine all terms under a common denominator, which might be \( (ab + a + 1)(ba + b + 1)(ac + 2c + 2) \).

### Final expression:

The solution would depend on detailed working through these algebraic identities and reductions.

Calculating it explicitly is complex and often varies based on interpreted values of \(a, b, c\) etc., but given \(abc = 2\), you could substitute relevant values to find particular results based on \(a, b, c\).

For a concrete numerical answer, specific values of \(a, b, c\) should be attempted. Alternatively, numerical methods or symbolic tools could be employed for precise simplification within bounds defined by \(abc = 2\).