1. **State the problem:** Solve the quadratic equation by factoring: $$6x^2 - 5x + 6 = 6x + 3$$ 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero: $$6x^2 - 5x + 6 - 6x - 3 = 0$$ Simplify the left side: $$6x^2 - 11x + 3 = 0$$ 3. **Use the factoring method:** We want to factor the quadratic expression: $$6x^2 - 11x + 3$$ We look for two numbers that multiply to $6 \times 3 = 18$ and add to $-11$. These numbers are $-9$ and $-2$ because $-9 \times -2 = 18$ and $-9 + -2 = -11$. 4. **Rewrite the middle term using these numbers:** $$6x^2 - 9x - 2x + 3 = 0$$ 5. **Group terms and factor each group:** $$3x(2x - 3) - 1(2x - 3) = 0$$ 6. **Factor out the common binomial:** $$(3x - 1)(2x - 3) = 0$$ 7. **Set each factor equal to zero and solve for $x$:** $$3x - 1 = 0 \implies 3x = 1 \implies x = \frac{1}{3}$$ $$2x - 3 = 0 \implies 2x = 3 \implies x = \frac{3}{2}$$ **Final answer:** $$x = \frac{1}{3} \text{ or } x = \frac{3}{2}$$