$f(x)=x^{2}+3 x-5$

Step 1 ：Understand the problem: We are given a quadratic function \(f(x) = x^{2} + 3x - 5\). There is no specific question asked about this function, so we will find its roots as a basic analysis.

Step 2 ：Set the function equal to zero to find the roots: \(x^{2} + 3x - 5 = 0\).

Step 3 ：Use the quadratic formula to solve for \(x\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^{2} + bx + c = 0\). In this case, \(a = 1\), \(b = 3\), and \(c = -5\).

Step 4 ：Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-3 \pm \sqrt{3^{2} - 4(1)(-5)}}{2(1)}\).

Step 5 ：Simplify the expression under the square root: \(x = \frac{-3 \pm \sqrt{9 + 20}}{2}\).

Step 6 ：Further simplify the expression: \(x = \frac{-3 \pm \sqrt{29}}{2}\).

Step 7 ：So, the roots of the function \(f(x) = x^{2} + 3x - 5\) are \(x = \frac{-3 + \sqrt{29}}{2}\) and \(x = \frac{-3 - \sqrt{29}}{2}\).

Step 8 ：Check the results: Substitute the roots back into the original equation to verify that they satisfy the equation. Both roots satisfy the equation, so the solution is correct.

Step 9 ：The final answer is \(x = \frac{-3 + \sqrt{29}}{2}\) and \(x = \frac{-3 - \sqrt{29}}{2}\).