Operations with rational expressions

A rational expression is a fraction with polynomials top and bottom, and the rules are the ones you already use with numerical fractions.

To simplify, factor the numerator and denominator completely, then cancel any factor they share. The crucial restriction is that you may cancel a factor but never a single term: in $\frac{x+2}{x}$ the $x$ does not cancel, because the top is a sum rather than a product.

To add or subtract, you need a common denominator exactly as with numbers. To multiply, multiply across and then cancel. To divide, multiply by the reciprocal.

Any value making a denominator zero is excluded, because the expression is undefined there — and that stays true even after cancelling. In $\frac{(x+2)(x-3)}{x+2}$, the simplified form is $x - 3$, but $x = -2$ is still not allowed.

Simplify $\frac{(x+2)(x-3)}{x+2}$.
The bracket $(x+2)$ is a common factor, so it cancels

Can the $x$ in $\frac{x+5}{x}$ be cancelled?
No. The numerator is a sum, not a product, so $x$ is not a common factor. Only factors cancel.