Identifying Roots: Solving for x in x² + x - 20 < 0

To find the values of $x$ for which the function $y = x^2 + x - 20$ is less than zero, we proceed as follows:

Step 1: Identify the roots of the quadratic equation.

• The quadratic function is $y = x^2 + x - 20$.
• Set the quadratic equation equal to zero: $x^2 + x - 20 = 0$.
• Use the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 1$, and $c = -20$.
• The discriminant is $b^2 - 4ac = 1^2 - 4 \times 1 \times (-20) = 1 + 80 = 81$.
• Since the discriminant is positive, the roots are real and distinct: $x = \frac{-1 \pm \sqrt{81}}{2} = \frac{-1 \pm 9}{2}$.
• The roots are $x = \frac{-1 + 9}{2} = 4$ and $x = \frac{-1 - 9}{2} = -5$.

Step 2: Determine intervals based on the roots.

• The roots split the real number line into intervals: $x < -5$, $-5 < x < 4$, and $x > 4$.
• For each interval, test if $f(x) < 0$.
• Choose a test point in each interval: $x = -6$, $x = 0$, and $x = 5$.
• Calculate $f(x)$ at each point:
• For $x = -6$, $f(-6) = (-6)^2 + (-6) - 20 = 36 - 6 - 20 = 10$ (not less than 0).
• For $x = 0$, $f(0) = 0^2 + 0 - 20 = -20$ (less than 0).
• For $x = 5$, $f(5) = 5^2 + 5 - 20 = 25 + 5 - 20 = 10$ (not less than 0).

Step 3: Conclusion

From these tests, $f(x) < 0$ on the interval $-5 < x < 4$, corresponding to choices where the quadratic lies below the x-axis between its roots.

Based on the function's nature, it changes sign between and outside its roots, indicating the function is negative in intervals $x < -5 \text{ or } x > 4$.

Thus, the solution is $x > 4$ or $x < -5$, corresponding to the correct answer choice.