The given expression is:

\[
-3x^2 - 12x - 17
\]

If you're looking to factor it or solve for \( x \), let's first check if it's factorable using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:

- \( a = -3 \)
- \( b = -12 \)
- \( c = -17 \)

Now let's compute the discriminant \( b^2 - 4ac \).

\[
\Delta = (-12)^2 - 4(-3)(-17) = 144 - 204 = -60
\]

Since the discriminant is negative (\( \Delta = -60 \)), the equation does not have real roots. It has two complex roots.

Now, solving for the roots:

\[
x = \frac{-(-12) \pm \sqrt{-60}}{2(-3)}
\]

\[
x = \frac{12 \pm \sqrt{60}i}{-6}
\]

Breaking it down:

\[
x = -2 \pm \frac{\sqrt{60}}{6} i
\]

This simplifies to:

\[
x = -2 \pm \frac{\sqrt{15}}{3} i
\]

Thus, the solutions are:

\[
x = -2 + \frac{\sqrt{15}}{3} i \quad \text{and} \quad x = -2 - \frac{\sqrt{15}}{3} i
\]

Would you like further details on this solution or any questions?

Here are 5 related questions to expand your understanding:

1. How can we simplify quadratic equations with complex roots?
2. How does the discriminant help determine the nature of the roots?
3. What is the significance of imaginary numbers in quadratic equations?
4. Can you show the steps for solving quadratics with a real discriminant?
5. What happens when the discriminant equals zero?

**Tip:** A negative discriminant always indicates the presence of complex roots.

Learn how to solve the quadratic equation -3x^2 - 12x - 17 using the quadratic formula. The discriminant shows complex roots for this equation, and the step-by-step solution simplifies the process. Suitable for high school students (Grades 9-12) studying quadratic equations and complex numbers.

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