Simplifying expressions

Simplifying an expression means rewriting it in the cleanest equal form, using whatever tool fits — combining like terms, factoring, or cancelling.

The most basic move is to combine like terms after expanding.
For $3(x + 2) - 2(x - 1)$, expand to $3x + 6 - 2x + 2$, then combine to get $x + 8$.
Watch the signs when a minus sits in front of a bracket.

Another move is to factor out a common factor, which is especially handy before solving an equation.
In $-2x^2 + 4x + 6 = 0$, every term shares a factor of $-2$, so pull it out: $-2(x^2 - 2x - 3) = 0$.
Dividing by $-2$ leaves the much simpler $x^2 - 2x - 3 = 0$, which factors as $(x - 3)(x + 1) = 0$.

With a rational expression, factor the top and bottom and cancel what they share.
For $\frac{x^2 + x - 6}{x^2 - 4}$, factor to $\frac{(x + 3)(x - 2)}{(x + 2)(x - 2)}$, then cancel the common $(x - 2)$ to leave $\frac{x + 3}{x + 2}$.
Whatever the tool, the goal is an equivalent expression that is simpler to read, solve, or compare.

Factor out the common factor to simplify $5x^2 - 10x$.
Both terms share $5x$, so pull it out: $5x(x - 2)$.
So $5x^2 - 10x = 5x(x - 2)$.

Simplify the rational expression $\frac{x^2 - 16}{x^2 + 2x - 8}$.
Factor top and bottom: $\frac{(x - 4)(x + 4)}{(x + 4)(x - 2)}$.
Cancel the common $(x + 4)$ to get $\frac{x - 4}{x - 2}$.

Simplify $4(x - 3) - 2(2x + 1)$.
Expand: $4x - 12 - 4x - 2$.
The $x$-terms cancel, leaving $-14$.