Rearrange the equation into the standard quadratic form ax² + 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:

$$3x^2 - 10x + 1 - 21 = 0$$ $$3x^2 - 10x - 20 = 0$$

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:

$$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(-20)}}{2(3)}$$ $$x = \frac{10 \pm \sqrt{100 - (-240)}}{6}$$ $$x = \frac{10 \pm \sqrt{100 + 240}}{6}$$ $$x = \frac{10 \pm \sqrt{340}}{6}$$

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:

$$x_1 = \frac{5 + \sqrt{85}}{3}$$ $$x_2 = \frac{5 - \sqrt{85}}{3}$$