It looks like you entered the equation:

\[ x^3 - 5x^2 + 12x + 6 = 0 \]

This is a cubic equation. To solve it, we can either try factoring it, use synthetic division, or apply methods like Cardano's formula. Let's start by checking if the equation can be factored or if there's an obvious root.

### Step 1: Check for Rational Roots
We can apply the Rational Root Theorem to find potential rational roots. The theorem states that any rational root, \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term (6) and \( q \) is a factor of the leading coefficient (1).

Possible values for \( p \) (factors of 6) are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Possible values for \( q \) (factors of 1) are \( \pm 1 \).

Thus, possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

### Step 2: Test Potential Roots
Let's test these values by substituting them into the equation.

1. **For \( x = 1 \):**

   \[
   1^3 - 5(1)^2 + 12(1) + 6 = 1 - 5 + 12 + 6 = 14 \quad (\text{not a root})
   \]

2. **For \( x = -1 \):**

   \[
   (-1)^3 - 5(-1)^2 + 12(-1) + 6 = -1 - 5 - 12 + 6 = -12 \quad (\text{not a root})
   \]

3. **For \( x = 2 \):**

   \[
   2^3 - 5(2)^2 + 12(2) + 6 = 8 - 20 + 24 + 6 = 18 \quad (\text{not a root})
   \]

4. **For \( x = -2 \):**

   \[
   (-2)^3 - 5(-2)^2 + 12(-2) + 6 = -8 - 20 - 24 + 6 = -46 \quad (\text{not a root})
   \]

5. **For \( x = 3 \):**

   \[
   3^3 - 5(3)^2 + 12(3) + 6 = 27 - 45 + 36 + 6 = 24 \quad (\text{not a root})
   \]

6. **For \( x = -3 \):**

   \[
   (-3)^3 - 5(-3)^2 + 12(-3) + 6 = -27 - 45 - 36 + 6 = -102 \quad (\text{not a root})
   \]

7. **For \( x = 6 \):**

   \[
   6^3 - 5(6)^2 + 12(6) + 6 = 216 - 180 + 72 + 6 = 114 \quad (\text{not a root})
   \]

8. **For \( x = -6 \):**

   \[
   (-6)^3 - 5(-6)^2 + 12(-6) + 6 = -216 - 180 - 72 + 6 = -462 \quad (\text{not a root})
   \]

Since none of the rational roots work, the equation might not have a simple rational root. We might need to use numerical methods or Cardano's formula to find the roots.

Would you like to proceed with numerical approximation, or would you prefer a more detailed exploration using Cardano's method?

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**Related Questions:**
1. What is the Rational Root Theorem, and how is it used to find possible roots?
2. How does Cardano's method work for solving cubic equations?
3. What are synthetic division and how can it help in solving cubic equations?
4. Can all cubic equations be solved by factoring?
5. How are complex roots determined in cubic equations?

**Tip:** When dealing with cubic equations, checking for simple rational roots can save time before moving to more complex methods.

Learn how to solve the cubic equation x^3 - 5x^2 + 12x + 6 = 0 using the Rational Root Theorem and other advanced methods. Explore step-by-step solutions and understand the application of Cardano's formula in finding roots. Suitable for advanced high school students.

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