Subtract $15 m n-22 m+2 n$ from $14 m n-12 m+7 n$.
To subtract the polynomial \(15mn - 22m + 2n\) from \(14mn - 12m + 7n\), we must set up the expression as \((14mn - 12m + 7n) - (15mn - 22m + 2n)\). In this expression, \(m\) and \(n\) act as variables. We identify the terms involving \(mn\), \(m\), and \(n\) separately to perform the subtraction systematically.
When we subtract an entire polynomial, we must distribute the negative sign to each term inside the parentheses. The subtraction becomes:
\[(14mn - 12m + 7n) - 15mn - (-22m) - (2n)\]
Applying the rule that subtracting a negative is the same as adding a positive, we rewrite this as:
\[14mn - 12m + 7n - 15mn + 22m - 2n\]
Now, we group the like terms together to simplify the expression:
\[(14mn - 15mn) + (-12m + 22m) + (7n - 2n)\]
We calculate each group:
Combining these results, we get \(-mn + 10m + 5n\).
Comparing our result \(-mn + 10m + 5n\) to the given options, we see that it matches the first option. Thus, the correct choice is \(-mn + 10m + 5n\). The final answer is \(\boxed{-mn + 10m + 5n}\).
In real-life scenarios, such as accounting or stock management, you might need to find the net change in a specific item count. For instance, if a company had \(14mn - 12m + 7n\) items of a certain type in inventory and then removed \(15mn - 22m + 2n\) items, the algebraic subtraction performed here tells you exactly how the inventory count changed, resulting in \(-mn + 10m + 5n\). This type of calculation is common in optimization problems where resources are adjusted dynamically.