This is one way you could have your steps written:
[tex]\frac{2}{3-\sqrt{3x^2}}\\\\\\\frac{2(3+\sqrt{3x^2})}{(3-\sqrt{3x^2})((3+\sqrt{3x^2})}\\\\\\\frac{6+2\sqrt{3x^2}}{3^2 - \left(\sqrt{3x^2}\right)^2}\\\\\\\frac{6+2\sqrt{3}\sqrt{x^2}}{9 - 3x^2}\\\\\\\frac{6+2(\sqrt{3})x}{9 - 3x^2}\\\\\\\frac{6+2x\sqrt{3}}{9 - 3x^2}\\\\\\[/tex]
Explanation:
You have the right idea to multiply top and bottom by [tex]3+\sqrt{3x^2}[/tex] to get rid of the root in the denominator (using the difference of squares rule). Afterwards, we use the rules that [tex]\sqrt{x*y} = \sqrt{x}*\sqrt{y}[/tex] and [tex]\sqrt{x^2} = x, \ \text{ when } x \ge 0[/tex], to simplify [tex]2\sqrt{3x^2}[/tex] into [tex]2x\sqrt{3}[/tex]. This is where the 2x comes from.
