Multiplying and dividing a polynomial by a monomial
A term outside a bracket, or underneath a whole polynomial, applies to every term inside it separately, never just the first one.
A worked example
To expand 5x(x + 4), 5x multiplies the x and multiplies the 4 separately, giving 5x squared plus 20x, not 5x squared plus 4.
The mistake people actually make
The same one-term-at-a-time rule runs both ways: multiplying out a bracket and dividing every term of a polynomial by a monomial are the same distribution, forwards and backwards.