Let's solve the equation step by step:

\[
3x^2 - 5x = 6 - 2x
\]

### Step 1: Move all terms to one side
First, let's move all the terms to the left side of the equation:

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

Simplify the like terms:

\[
3x^2 - 3x - 6 = 0
\]

### Step 2: Factor the quadratic equation
We have:

\[
3x^2 - 3x - 6 = 0
\]

We can factor out a 3 from all the terms:

\[
3(x^2 - x - 2) = 0
\]

Now, we can factor \( x^2 - x - 2 \). The factors of -2 that add up to -1 are -2 and 1. So, we can factor this as:

\[
3(x - 2)(x + 1) = 0
\]

### Step 3: Solve for \( x \)
Now, set each factor equal to zero:

\[
x - 2 = 0 \quad \text{or} \quad x + 1 = 0
\]

Solving these gives:

\[
x = 2 \quad \text{or} \quad x = -1
\]

### Final Answer:
The solutions are:

\[
x = 2 \quad \text{or} \quad x = -1
\]

Would you like further details or explanations on any part of the process?

Here are some related questions:
1. What are other methods to solve quadratic equations?
2. Can you explain how to use the quadratic formula for this problem?
3. What happens if a quadratic equation has no real solutions?
4. How can you verify the solutions to this equation?
5. How does completing the square work for this kind of equation?

**Tip:** Always check your factored equation by expanding it back to ensure the factorization is correct.

Learn how to solve the quadratic equation 3x^2 - 5x = 6 - 2x step-by-step by factoring and using the zero product property. This solution simplifies the equation and shows how to find the roots. Ideal for students in Grades 9-10 studying quadratic equations.

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