Solve the equation using the quadratic formula. \[ x^{2}+5 x+3=0 \] The solution set is (Simplify your answer. Type an exact answer, using radicals and \( i \) as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

To solve the quadratic equation \( x^{2}+5x+3=0 \) using the quadratic formula, we start by identifying the coefficients: \( a = 1 \), \( b = 5 \), and \( c = 3 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] First, we calculate the discriminant: \[ b^2 - 4ac = 5^2 - 4(1)(3) = 25 - 12 = 13 \] Now, substituting into the quadratic formula: \[ x = \frac{-5 \pm \sqrt{13}}{2(1)} = \frac{-5 \pm \sqrt{13}}{2} \] Thus, the solution set is: \[ \boxed{\frac{-5 + \sqrt{13}}{2}, \frac{-5 - \sqrt{13}}{2}} \]