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The image shows a quadratic equation that needs to be solved by factoring. The equation given is:

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

Let's solve this step by step.

### Step 1: Simplify the equation
First, we need to bring all terms to one side of the equation:

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

Simplifying the above expression:

\[ 6x^2 - 11x + 3 = 0 \]

### Step 2: Factor the quadratic equation
Now, we need to factor the quadratic equation \( 6x^2 - 11x + 3 = 0 \).

To factor it, we look for two numbers that multiply to \( 6 \times 3 = 18 \) and add to \( -11 \).

The numbers that work are \( -9 \) and \( -2 \), so we rewrite the middle term using these numbers:

\[ 6x^2 - 9x - 2x + 3 = 0 \]

### Step 3: Group the terms
Group the terms to factor by grouping:

\[ (6x^2 - 9x) - (2x - 3) = 0 \]

Factor out the common factors from each group:

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

### Step 4: Factor out the common binomial
Now factor out the common binomial factor:

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

### Step 5: Solve for \( x \)
Set each factor equal to zero and solve for \( x \):

1. \( 2x - 3 = 0 \)

   \[
   2x = 3 \quad \Rightarrow \quad x = \frac{3}{2}
   \]

2. \( 3x - 1 = 0 \)

   \[
   3x = 1 \quad \Rightarrow \quad x = \frac{1}{3}
   \]

### Final Answer:
The solutions are \( x = \frac{3}{2} \) and \( x = \frac{1}{3} \).

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Would you like any further details on this solution? Here are some related questions that might interest you:

1. How do you factor quadratic equations with coefficients larger than 1?
2. What is the significance of the discriminant in solving quadratic equations?
3. Can all quadratic equations be factored, and if not, why?
4. How does completing the square compare to factoring when solving quadratics?
5. How can you check if your factored solution is correct?

**Tip:** When factoring quadratic equations, always double-check by expanding the factors to ensure they match the original equation.

Solve the quadratic by factoring: 6x^2 - 5x + 6 = 6x + 3

This page provides a step-by-step solution to solving the quadratic equation 6x^2 - 5x + 6 = 6x + 3 by factoring. Learn the core algebraic concepts and techniques used, suitable for students in Grades 9-11.

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