The equation shown in the image is:

$-3x^3 + 4x^2 + x + 6 = 0$

This is a cubic equation, and solving it involves finding the roots (values of $x$) that satisfy the equation. Let me guide you through the process of solving it.

Step 1: Check for Rational Roots Using the Rational Root Theorem

The Rational Root Theorem suggests that the possible rational roots are factors of the constant term (6) divided by factors of the leading coefficient (-3). These potential roots are:

$\pm 1, \pm 2, \pm 3, \pm 6$

We will substitute these values into the equation to check if any are roots.

Step 2: Test Potential Roots

I will now test the possible values of $x$.The value $x = 2$ is a root of the equation. Now that we have found a root, we can factor the cubic polynomial by dividing it by $(x - 2)$.

Step 3: Polynomial Division

We will divide the cubic equation by $(x - 2)$ to reduce it to a quadratic equation. Afterward, we can solve the quadratic part.

Let me perform the polynomial division.The result of dividing $-3x^3 + 4x^2 + x + 6$ by $(x - 2)$ is:

$-3x^2 - 2x - 3$

This is the quadratic factor. Now, we can solve the quadratic equation $-3x^2 - 2x - 3 = 0$.

Step 4: Solve the Quadratic Equation

We will now solve the quadratic equation for its roots.The quadratic equation $-3x^2 - 2x - 3 = 0$ has two complex roots:

$x = -\frac{1}{3} - \frac{2\sqrt{2}}{3}i \quad \text{and} \quad x = -\frac{1}{3} + \frac{2\sqrt{2}}{3}i$

Summary of Roots:

• Real root: $x = 2$
• Complex roots: $x = -\frac{1}{3} \pm \frac{2\sqrt{2}}{3}i$

Would you like a more detailed explanation of any of these steps, or have further questions?

Here are five related questions for further exploration:

1. How do you perform synthetic division manually?
2. What is the Rational Root Theorem, and how does it help in solving polynomials?
3. How are complex roots derived from quadratic equations?
4. Can all cubic equations have both real and complex roots?
5. What are the general methods to solve polynomials of degree higher than 2?

Tip: When solving higher-degree polynomials, always check for possible rational roots first, as they can simplify the equation significantly.