To find the domain of the function \( f(x) = \frac{2x + 1}{x^2 + x - 20} \), we need to determine the values of \( x \) for which the function is defined. The function will be undefined wherever the denominator is zero.
First, let's find the values for which the denominator \( x^2 + x - 20 \) is zero. We can factor the quadratic expression:
\[ x^2 + x - 20 = (x - 4)(x + 5) \]
Now, set the factored form equal to zero:
\[ (x - 4)(x + 5) = 0 \]
This gives us two solutions:
\[ x - 4 = 0 \implies x = 4 \] \[ x + 5 = 0 \implies x = -5 \]
Thus, the function \( f(x) \) is undefined at \( x = 4 \) and \( x = -5 \).
To find the domain, we need to exclude these values from the set of all real numbers. Therefore, the domain in interval notation is:
\[ (-\infty, -5) \cup (-5, 4) \cup (4, \infty) \]
In summary, the domain of the function \( f(x) \) is
\[ (-\infty, -5) \cup (-5, 4) \cup (4, \infty). \]