Answer:
The expression is given below as
[tex]\frac{4}{\sqrt{x-2}-\sqrt{x}}[/tex]
Concept:
To rationalize the denominator, we will multiply by the conjugate given below
The conjugate is given below as
[tex]\frac{\sqrt{x-2}+\sqrt{x}}{\sqrt{x-2}+\sqrt{x}}[/tex]
Step 1:
Multiply the expression in the question by the conjugate, we will have
[tex]\frac{4}{\sqrt{x-2}-\sqrt{x}}\times\frac{\sqrt{x-2}+\sqrt{x}}{\sqrt{x-2}+\sqrt{x}}[/tex]
By expanding the brackets, we will have
[tex]\begin{gathered} \frac{4\sqrt{x-2}+4\sqrt{x}}{(\sqrt{x-2)^2-(\sqrt{x})^2}} \\ =\frac{4\sqrt{x-2}+4\sqrt{x}}{x-2-x} \\ =\frac{4\sqrt{x-2}+4\sqrt{x}}{-2} \end{gathered}[/tex]
Step 2:
Factor our the common number and divide
[tex]\begin{gathered} =\frac{4(\sqrt{x-2}+\sqrt{x})}{-2} \\ =-2(\sqrt{x}+\sqrt{x-2)} \end{gathered}[/tex]
Hence,
The final answer is
[tex]\Rightarrow-2(\sqrt{x}+\sqrt{x-2)}[/tex]
OPTION A is the right answer
