Which of the following equations is correct?

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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:

$$\sqrt{ab}=\sqrt{a}\textcolor{red}{\cdot}\sqrt{b}$$