Solve the function. \( \begin{array}{c}x^{3}+5 x^{2}-3 x-15=0 \\ x=[?], \pm \sqrt{[]} \\ \text { simplify your answer. }\end{array} \)

To solve the cubic equation \(x^{3}+5x^{2}-3x-15=0\), we can look for rational roots using the Rational Root Theorem or factorization if possible. Testing \(x = 3\): \[ 3^3 + 5(3)^2 - 3(3) - 15 = 27 + 45 - 9 - 15 = 48 \quad \text{(not a root)} \] Next, testing \(x = -3\): \[ (-3)^3 + 5(-3)^2 - 3(-3) - 15 = -27 + 45 + 9 - 15 = 12 \quad \text{(not a root)} \] After testing several integers, we can find \(x = -5\): \[ (-5)^3 + 5(-5)^2 - 3(-5) - 15 = -125 + 125 + 15 - 15 = 0 \quad \text{(this works!)} \] Now we can factor out \(x + 5\) from the cubic polynomial using synthetic division or polynomial long division. The remaining polynomial is \(x^2 - 3\). Thus, the factorization is: \[ x^3 + 5x^2 - 3x - 15 = (x + 5)(x^2 - 3) = 0 \] Setting each factor to zero gives us: 1. \(x + 5 = 0 \quad \Rightarrow \quad x = -5\) 2. \(x^2 - 3 = 0 \quad \Rightarrow \quad x = \pm \sqrt{3}\) The final solutions are: \[ x = -5, \pm \sqrt{3} \] You can express these results as \(x = [-5], \pm \sqrt{[3]}\).