$$ 2 x^{2}+x+3=0 $$ The solution set is $$ \} $$. Use the quadratic formula to solve the equation. (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

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To solve the quadratic equation $$ 2x^{2} + x + 3 = 0 $$ using the quadratic formula, follow these steps:

The quadratic formula is:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
where $$ a = 2 $$, $$ b = 1 $$, and $$ c = 3 $$.

1. **Calculate the discriminant ($$ \Delta $$):**
$$
\Delta = b^2 - 4ac = 1^2 - 4 \cdot 2 \cdot 3 = 1 - 24 = -23
$$
Since the discriminant is negative, the solutions will be complex numbers.

2. **Substitute the values into the formula:**
$$
x = \frac{-1 \pm \sqrt{-23}}{2 \cdot 2} = \frac{-1 \pm i\sqrt{23}}{4}
$$
where $$ i $$ is the imaginary unit.

3. **Write the solution set:**
$$
\left\{ \frac{-1 + i\sqrt{23}}{4}, \; \frac{-1 - i\sqrt{23}}{4} \right\}
$$

**Final Answer:**
$$
\left\{\, \dfrac{-1 + \sqrt{-23}}{4},\; \dfrac{-1 - \sqrt{-23}}{4} \,\right\}
$$
The solutions are $$ x = \frac{-1 + i\sqrt{23}}{4} $$ and $$ x = \frac{-1 - i\sqrt{23}}{4} $$.

\( 2 x^{2}+x+3=0 \) The solution set is \( \} \). Use the quadratic formula to solve the equation. (Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)

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