Cancel with Care

In $\frac{4x^{2}+2x}{2x}$, you cannot cross out the $2x$ on top against the $2x$ below and leave the rest alone. The denominator divides the whole sum, not one term of it.

Divide each term by $2x$: $\frac{4x^{2}}{2x} = 2x$ and $\frac{2x}{2x} = 1$, for nonzero $x$. The quotient is $2x+1$. A nonzero expression divided by itself is 1, so the second term becomes 1. It does not disappear.

You may cancel a factor that is shared by the whole numerator and the whole denominator. You may not cancel one term inside a sum. If you factor the numerator as $2x\left(2x+1\right)$, then $2x$ is a factor of the whole top, and you can cancel it. Cancelling leaves $2x+1$, for nonzero $x$.