Impulse of finding values of individual variables

It is tempting to solve for each variable first and then plug in.
But two equations with squared terms usually cannot be solved neatly for each variable.
And guessing values is risky, because the variables might not be whole numbers.

The smarter move is to work directly with the expression the question asks for.
Write out what that expression expands to, then match it against the equations you were given.
Often a simple sum or difference of the equations produces exactly what you need.

You want $(m - n)^2$, and $(m - n)^2 = m^2 - 2mn + n^2$.
Adding $m^2 + 3mn = 5$ and $n^2 - 5mn = 12$ gives $m^2 + n^2 - 2mn = 17$.
That is exactly $(m - n)^2 = 17$, with no need to find $m$ or $n$.

This idea also applies to simpler problems.
If a question asks for $3 - 2x$ rather than $x$, you can often rearrange to get $3 - 2x$ directly.
Always check what the question actually asks for before you start solving.

Given $a^2 + 2ab = 3$ and $b^2 - 4ab = 5$, find $(a - b)^2$.
Note that $(a - b)^2 = a^2 - 2ab + b^2$.
Add the two equations: $a^2 + b^2 - 2ab = 8$, so $(a - b)^2 = 8$.

Given $m^2 + mn = 2$ and $n^2 - 3mn = 4$, find $2(m - n)^2$.
Adding the equations gives $m^2 + n^2 - 2mn = 6$, which is $(m - n)^2 = 6$.
So $2(m - n)^2 = 12$.

A question gives an equation and asks for $3 - 2x$, not $x$. Should you solve for $x$ first?
Not necessarily.
If you can rearrange the equation to isolate $3 - 2x$ directly, you save time and avoid extra work.