Step 1: Rearrange the equation into the standard quadratic form $ax^2 + bx + c = 0$. The given equation is: $$3x^2 + 2x + 1 = 12x + 21$$ Subtract $12x$ from both sides: $$3x^2 + 2x - 12x + 1 = 21$$ $$3x^2 - 10x + 1 = 21$$ Subtract $21$ from both sides: $$3x^2 - 10x + 1 - 21 = 0$$ $$3x^2 - 10x - 20 = 0$$ Step 2: Solve the quadratic equation using the quadratic formula. The quadratic equation is in the form $ax^2 + bx + c = 0$, where $a=3$, $b=-10$, and $c=-20$. The quadratic formula is: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substitute the values of $a$, $b$, and $c$ into the formula: $$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(-20)}}{2(3)}$$ $$x = \frac{10 \pm \sqrt{100 - (-240)}}{6}$$ $$x = \frac{10 \pm \sqrt{100 + 240}}{6}$$ $$x = \frac{10 \pm \sqrt{340}}{6}$$ Step 3: Simplify the square root and the expression for $x$. Simplify $\sqrt{340}$: $$\sqrt{340} = \sqrt{4 \times 85} = \sqrt{4} \times \sqrt{85} = 2\sqrt{85}$$ Substitute this back into the expression for $x$: $$x = \frac{10 \pm 2\sqrt{85}}{6}$$ Factor out $2$ from the numerator: $$x = \frac{2(5 \pm \sqrt{85})}{6}$$ Divide the numerator and denominator by $2$: $$x = \frac{5 \pm \sqrt{85}}{3}$$ The two solutions for $x$ are: $$x_1 = \frac{5 + \sqrt{85}}{3}$$ $$x_2 = \frac{5 - \sqrt{85}}{3}$$ The final answer is $\boxed{x = \frac{5 \pm \sqrt{85}}{3}}$.