e) \( 3 x^{2}+4 x+1 \)

Did you know that the quadratic equation \(3x^2 + 4x + 1\) can be solved using the quadratic formula? The formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), allows us to find the values of \(x\) where the equation equals zero. Here, \(a = 3\), \(b = 4\), and \(c = 1\). Plugging in these values can lead you right to the solutions! When you graph this equation, you'll see it's a parabola that opens upwards, since the coefficient of \(x^2\) (which is 3) is positive. The vertex represents the minimum point of the parabola, and the axis of symmetry is found at \(x = -\frac{b}{2a}\). This visual representation can help you understand the behavior of the function better and locate the minimum value quite easily!