Find the the domain of the function \( f(x)=\frac{x+3}{x^{2}+10 x-11} \) \( \begin{array}{l}\{x \mid x \neq-11 \text { and } x \neq 1 \text { and } x \neq-3\} \\ \{x \mid x \neq-3\} \\ \{x \mid x \neq 11\} \\ \{x \mid x \neq-11 \text { and } x \neq 1\} \\ \{x \mid x \neq 11 \text { and } x \neq-1\}\end{array} \)

To find the domain of the function \( f(x) = \frac{x+3}{x^2 + 10x - 11} \), we need to determine where the function is defined. The function is undefined where the denominator is equal to zero. First, we need to find the roots of the denominator: \[ x^2 + 10x - 11 = 0 \] We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 10 \), and \( c = -11 \). Calculating the discriminant: \[ b^2 - 4ac = 10^2 - 4 \cdot 1 \cdot (-11) = 100 + 44 = 144 \] Now applying the quadratic formula: \[ x = \frac{-10 \pm \sqrt{144}}{2 \cdot 1} = \frac{-10 \pm 12}{2} \] This gives us two solutions: 1. \( x = \frac{-10 + 12}{2} = \frac{2}{2} = 1 \) 2. \( x = \frac{-10 - 12}{2} = \frac{-22}{2} = -11 \) Thus, the function \( f(x) \) is undefined at \( x = 1 \) and \( x = -11 \). The function is defined for all other real numbers. Therefore, the domain of \( f(x) \) can be expressed as: \[ \{ x \mid x \neq -11 \text{ and } x \neq 1 \} \] Now, let's check the provided options: 1. \( \{x \mid x \neq -11 \text{ and } x \neq 1 \text{ and } x \neq -3\} \) 2. \( \{x \mid x \neq -3\} \) 3. \( \{x \mid x \neq 11\} \) 4. \( \{x \mid x \neq -11 \text{ and } x \neq 1\} \) 5. \( \{x \mid x \neq 11 \text{ and } x \neq -1\} \) The correct option that matches our derived domain is: \[ \{x \mid x \neq -11 \text{ and } x \neq 1\} \] Thus, the answer is option 4.