The correct expansion for the first option:
$$ (a+b)^2=(a+b)(a+b) $$
$$ = a^2+ab+ab+b^2 $$
$$ = a^2 + \textcolor{red}{2ab}+b^2 $$
For the second option, we would only be able to cancel out the \(B\) if it were included in the first term, like the following:
$$\frac{\textcolor{red}{B}+AB}{B} $$
$$=\frac{\cancel{B}(1+A)}{\cancel{B}} $$
$$ = 1+A $$
Here are the correct steps for simplifying the third option:
$$ \frac{1/x}{a/x-b/x} =\frac{\frac{1}{x}}{\frac{a}{x}-\frac{b}{x}} = \frac{\frac{1}{x}}{\frac{1}{x}(a-b)} $$
$$ \frac{\cancel{\frac{1}{x}}}{\cancel{\frac{1}{x}}(a-b)} =\frac{1}{a-b} $$
Last option:
$$ \sqrt{ab}=\sqrt{a}\textcolor{red}{\cdot}\sqrt{b} $$