To find the roots of $f(x) = -2x^2 - 4x - 2$ by graphing, we need to identify where the parabola intersects the x-axis. The roots are the values of $x$ where $f(x) = 0$. Here's the process:

Steps:

1. Graph the quadratic equation:

   • The function $f(x) = -2x^2 - 4x - 2$ is a downward-opening parabola (since the coefficient of $x^2$ is negative).
   • The vertex form or standard form of this equation provides a symmetrical parabola around its vertex.

2. Find the vertex and axis of symmetry:

   • The axis of symmetry is $x = -\frac{b}{2a}$, where $a = -2$ and $b = -4$. $x = -\frac{-4}{2(-2)} = -1$
   • The vertex occurs at $x = -1$. Plugging this into the function: $f(-1) = -2(-1)^2 - 4(-1) - 2 = -2 + 4 - 2 = 0$ Thus, the vertex is $(-1, 0)$.

3. Determine the roots:

   • Since the vertex lies on the x-axis, $(-1, 0)$ is also a root.
   • The parabola is symmetrical around the axis of symmetry, so there is no second root because the parabola touches the x-axis at a single point.

Roots:

The root of $f(x) = -2x^2 - 4x - 2$ is: $x = -1$

Would you like me to create a graph of this function for visual confirmation?