Find the domain of the rational expression. $$ f(x)=\frac{2}{x^{2}-2 x-35} $$

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1. The domain of the function is all real numbers except where the denominator is zero. We start with the denominator:
   $$
   x^{2} - 2x - 35.
   $$

2. Set the denominator equal to zero:
   $$
   x^{2} - 2x - 35 = 0.
   $$

3. Factor the quadratic. We look for two numbers that multiply to $$-35$$ and add up to $$-2$$. The numbers $$-7$$ and $$5$$ work because:
   $$
   -7 \times 5 = -35 \quad \text{and} \quad -7 + 5 = -2.
   $$
   Therefore, the factored form is:
   $$
   (x-7)(x+5) = 0.
   $$

4. Solve for $$ x $$ by setting each factor equal to zero:
   $$
   x - 7 = 0 \quad \Longrightarrow \quad x = 7,
   $$
   $$
   x + 5 = 0 \quad \Longrightarrow \quad x = -5.
   $$

5. Since the denominator cannot be zero, the values $$ x = 7 $$ and $$ x = -5 $$ must be excluded from the domain.

6. Thus, the domain of $$ f(x) $$ is:
   $$
   \{ x \in \mathbb{R} \mid x \neq -5 \text{ and } x \neq 7 \}.
   $$
The domain of $$ f(x) = \frac{2}{x^{2} - 2x - 35} $$ is all real numbers except $$ x = -5 $$ and $$ x = 7 $$.

Find the domain of the rational expression. \( f(x)=\frac{2}{x^{2}-2 x-35} \)

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