The expression in the image is:

$2 + \frac{3m - n}{5} - \frac{1 - m}{2}$

To simplify or manipulate this expression, we can:

1. Find a common denominator between the fractions. The denominators are 5 and 2, so the least common denominator (LCD) is 10.
2. Rewrite each term with the denominator 10: $2 + \frac{2(3m - n)}{10} - \frac{5(1 - m)}{10}$ This becomes: $2 + \frac{6m - 2n}{10} - \frac{5 - 5m}{10}$
3. Combine the fractions: $2 + \frac{(6m - 2n) - (5 - 5m)}{10}$ Simplify the numerator: $2 + \frac{6m - 2n - 5 + 5m}{10} = 2 + \frac{11m - 2n - 5}{10}$
4. The simplified expression is: $2 + \frac{11m - 2n - 5}{10}$ This can be rewritten as: $\frac{20}{10} + \frac{11m - 2n - 5}{10} = \frac{20 + 11m - 2n - 5}{10} = \frac{11m - 2n + 15}{10}$

Thus, the final simplified expression is: $\frac{11m - 2n + 15}{10}$

Would you like more details on any step or have any questions?

Here are some related questions to expand on this:

1. How can we solve this expression for $m$?
2. How can we solve this expression for $n$?
3. What happens if $m$ and $n$ are given specific values?
4. How would you handle this expression if additional terms were added?
5. Can this expression be factored further for certain values of $m$ and $n$?

Tip: Always look for the least common denominator (LCD) when working with fractions—it helps to combine terms efficiently!