To simplify [tex]\(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}\)[/tex], let's go through the steps shown in the work:
1. Start with the original expression:
[tex]\[
\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}
\][/tex]
2. Multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}} \times \frac{\sqrt{3}+\sqrt{x}}{\sqrt{3}+\sqrt{x}} = \frac{\sqrt{3}(\sqrt{3}+\sqrt{x})}{(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x})}
\][/tex]
3. Simplify the numerator:
[tex]\[
\sqrt{3}(\sqrt{3}+\sqrt{x}) = \sqrt{3}\cdot\sqrt{3} + \sqrt{3}\cdot\sqrt{x} = 3 + \sqrt{3x}
\][/tex]
4. Simplify the denominator using the difference of squares:
[tex]\[
(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x}) = (\sqrt{3})^2 - (\sqrt{x})^2 = 3 - x
\][/tex]
5. Combine the simplified numerator and denominator:
[tex]\[
\frac{3 + \sqrt{3x}}{3 - x}
\][/tex]
So, the simplest form of [tex]\(\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}\)[/tex] is:
[tex]\[
\frac{3 + \sqrt{3x}}{3 - x}
\][/tex]
Among the given options, this corresponds to:
[tex]\[
\boxed{\frac{3+\sqrt{3 x}}{3-x}}
\][/tex]
