Write the given expression in the form \( f(x)=a(x-h)^{2}+k \). Identify the vertex. \( f(x)=2-15 x-3 x^{2} \)

To rewrite the expression \( f(x) = 2 - 15x - 3x^2 \) in vertex form, we first rearrange it into standard form by factoring out \(-3\) from the \(x^2\) term and the linear term: \[ f(x) = -3(x^2 + 5x) + 2 \] Next, we need to complete the square inside the parentheses. We take half of the coefficient of \(x\), which is \(5\), divide it by \(2\), and square it: \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] Now, we can add and subtract \(\frac{25}{4}\) inside the parentheses: \[ f(x) = -3\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) + 2 \] \[ = -3\left((x + \frac{5}{2})^2 - \frac{25}{4}\right) + 2 \] \[ = -3(x + \frac{5}{2})^2 + \frac{75}{4} + 2 \] \[ = -3(x + \frac{5}{2})^2 + \frac{75}{4} + \frac{8}{4} \] \[ = -3(x + \frac{5}{2})^2 + \frac{83}{4} \] Thus, the equation in vertex form is: \[ f(x) = -3\left(x + \frac{5}{2}\right)^{2} + \frac{83}{4} \] The vertex of the parabola represented by this function is the point \((h, k)\), which is: \[ \left(-\frac{5}{2}, \frac{83}{4}\right) \]