Identify the domain of the expression. $$ \frac{5 x+13}{x^{2}-6 x-7} $$ The domain is all real numbers except

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To find the domain of the expression $$ \frac{5x + 13}{x^2 - 6x - 7} $$, we need to identify the values of $$ x $$ that make the denominator equal to zero, as these values will not be included in the domain.

1. **Set the denominator equal to zero:**
   $$
   x^2 - 6x - 7 = 0
   $$

2. **Factor the quadratic equation:**
   We need to find two numbers that multiply to $$-7$$ (the constant term) and add to $$-6$$ (the coefficient of $$ x $$). The numbers $$-7$$ and $$1$$ satisfy these conditions.

Thus, we can factor the equation as:
   $$
   (x - 7)(x + 1) = 0
   $$

3. **Solve for $$ x $$:**
   Setting each factor equal to zero gives us:
   $$
   x - 7 = 0 \quad \Rightarrow \quad x = 7
   $$
   $$
   x + 1 = 0 \quad \Rightarrow \quad x = -1
   $$

4. **Identify the values to exclude from the domain:**
   The values $$ x = 7 $$ and $$ x = -1 $$ make the denominator zero, so these values must be excluded from the domain.

5. **State the domain:**
   The domain of the expression is all real numbers except $$ x = 7 $$ and $$ x = -1 $$.

Thus, the domain is:
$$
\text{All real numbers except } x = 7 \text{ and } x = -1.
$$
The domain is all real numbers except $$ x = 7 $$ and $$ x = -1 $$.

Identify the domain of the expression. \( \frac{5 x+13}{x^{2}-6 x-7} \) The domain is all real numbers except

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