1. **Problem:** Solve the equation $$3x^2 + 2x - \sqrt{3x^2 + 2x - 1} = 3$$. 2. **Step 1: Isolate the square root term.** Move the square root term to the right side: $$3x^2 + 2x - 3 = \sqrt{3x^2 + 2x - 1}$$ 3. **Step 2: Square both sides to eliminate the square root.** $$\left(3x^2 + 2x - 3\right)^2 = 3x^2 + 2x - 1$$ 4. **Step 3: Expand the left side.** $$\left(3x^2 + 2x - 3\right)^2 = (3x^2)^2 + 2 \cdot 3x^2 \cdot 2x + (2x)^2 - 2 \cdot 3x^2 \cdot 3 - 2 \cdot 2x \cdot 3 + 3^2$$ Calculate stepwise: $$= 9x^4 + 12x^3 + 4x^2 - 18x^2 - 12x + 9$$ Simplify terms: $$= 9x^4 + 12x^3 - 14x^2 - 12x + 9$$ 5. **Step 4: Set up the equation.** $$9x^4 + 12x^3 - 14x^2 - 12x + 9 = 3x^2 + 2x - 1$$ 6. **Step 5: Bring all terms to one side.** $$9x^4 + 12x^3 - 14x^2 - 12x + 9 - 3x^2 - 2x + 1 = 0$$ Simplify: $$9x^4 + 12x^3 - 17x^2 - 14x + 10 = 0$$ 7. **Step 6: Solve the quartic equation.** This quartic is complex; try rational root theorem for possible roots among factors of 10 over 9: $$\pm1, \pm2, \pm5, \pm10, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{5}{3}, \pm\frac{10}{3}, \pm\frac{1}{9}, \pm\frac{2}{9}, \pm\frac{5}{9}, \pm\frac{10}{9}$$ Test $x=1$: $$9(1)^4 + 12(1)^3 - 17(1)^2 - 14(1) + 10 = 9 + 12 - 17 - 14 + 10 = 0$$ So, $x=1$ is a root. 8. **Step 7: Polynomial division to factor out $(x-1)$.** Divide $$9x^4 + 12x^3 - 17x^2 - 14x + 10$$ by $(x-1)$: Quotient: $$9x^3 + 21x^2 + 4x - 10$$ 9. **Step 8: Solve cubic $$9x^3 + 21x^2 + 4x - 10 = 0$$.** Try rational roots again: test $x=1$: $$9 + 21 + 4 - 10 = 24 \neq 0$$ Test $x=\frac{1}{3}$: $$9(\frac{1}{3})^3 + 21(\frac{1}{3})^2 + 4(\frac{1}{3}) - 10 = 9(\frac{1}{27}) + 21(\frac{1}{9}) + \frac{4}{3} - 10 = \frac{1}{3} + \frac{7}{3} + \frac{4}{3} - 10 = 4 - 10 = -6 \neq 0$$ Try $x=-2$: $$9(-2)^3 + 21(-2)^2 + 4(-2) - 10 = 9(-8) + 21(4) - 8 - 10 = -72 + 84 - 8 - 10 = -6 \neq 0$$ No easy rational roots; use numerical or approximate methods. 10. **Step 9: Check for extraneous solutions.** Recall original equation has a square root; solutions must satisfy the domain: $$3x^2 + 2x - 1 \geq 0$$ 11. **Step 10: Verify $x=1$ in original equation.** Calculate left side: $$3(1)^2 + 2(1) - \sqrt{3(1)^2 + 2(1) - 1} = 3 + 2 - \sqrt{3 + 2 - 1} = 5 - \sqrt{4} = 5 - 2 = 3$$ Right side is 3, so $x=1$ is a valid solution. **Final answer:** $$\boxed{x=1}$$