Given that $b$ is rational and $i=\sqrt{-1}$, the product of the expression $(3+bi)$ and which of the following expressions must be a rational number?

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If you have time, you can try out all the options.

The correct option would be to multiply $(3+bi)$ with its conjugate so we can use difference of squares:

$$(a+b)(a-b) = a^2-b^2$$ $$(3+bi)(3-bi) = (3)^2-(bi)^2$$ $$= 9 -b^2i^2$$ $$= 9+b^2$$