To solve for the discriminant of the given quadratic equation $f(x) = -3x^2 - 2x - 1$, we use the formula for the discriminant:

$\Delta = b^2 - 4ac$

Step-by-Step Calculation:

1. Identify the coefficients from $f(x) = -3x^2 - 2x - 1$:

   • $a = -3$
   • $b = -2$
   • $c = -1$

2. Substitute these values into the discriminant formula: $\Delta = (-2)^2 - 4(-3)(-1)$

3. Simplify: $\Delta = 4 - 4(3)(1)$ $\Delta = 4 - 12$ $\Delta = -8$

Final Answer:

The discriminant is $\Delta = -8$, so the correct option is B.

Would you like further clarification or a deeper explanation of how discriminants determine the nature of roots?

Related Questions:

1. What does a negative discriminant imply about the roots of a quadratic equation?
2. How do you solve quadratic equations when the discriminant is negative?
3. Can the discriminant formula be extended to higher-degree polynomials?
4. How does the discriminant relate to the vertex of a parabola?
5. Why is the discriminant important in understanding the behavior of a function?

Tip:

Remember, the sign of the discriminant ($\Delta$) determines the type of roots: positive ($2$ real roots), zero ($1$ real root), or negative ($2$ complex roots).