Reframing the question

Reframing means rewriting what you are given so it lines up with what you need to find.
Instead of grinding through a direct solution, you reshape the expression or equation.
A few standard moves cover most cases.

One move is to rewrite an expression so a needed piece appears.
To express $2x + 7$ in terms of $(x + 3)$, build from $2(x + 3) = 2x + 6$.
Then $2x + 7 = 2(x + 3) + 1$.

Another move is to treat a whole chunk as a single variable.
In $\frac{2x - 3}{-2} + 3 - 2x = 4$, let $m = 3 - 2x$, so $2x - 3 = -m$.
The equation becomes $\frac{-m}{-2} + m = 4$, which is far simpler.

You can also combine given equations to produce exactly what is asked.
To find $m^2 + n^2 - 2mn$ from $m^2 + 3mn = 5$ and $n^2 - 5mn = 12$, add them.
The result $m^2 + n^2 - 2mn = 17$ is the expression you wanted, without ever finding $m$ or $n$.

Rewrite $3x + 10$ in terms of $(x + 2)$. \ Since $3(x + 2) = 3x + 6$, that is $4$ less than $3x + 10$. \ So $3x + 10 = 3(x + 2) + 4$.

Given $a + 2b = 7$ and $a - b = 1$, find $2a + b$ without solving for $a$ and $b$. \ Add the two equations: $(a + 2b) + (a - b) = 7 + 1$. \ This gives $2a + b = 8$.

Solve $(x + 1)^2 - 5(x + 1) + 6 = 0$ by letting $u = x + 1$. \ The equation becomes $u^2 - 5u + 6 = 0$, which factors as $(u - 2)(u - 3) = 0$, so $u = 2$ or $u = 3$. \ Since $u = x + 1$, we get $x = 1$ or $x = 2$.