1. **State the problem:** Write the quadratic function $$f(x) = 4 - 9x - 3x^2$$ in vertex form $$f(x) = a(x - h)^2 + k$$ and identify the vertex. 2. **Rewrite the function in standard form:** Arrange terms by descending powers of $$x$$: $$f(x) = -3x^2 - 9x + 4$$ 3. **Factor out the coefficient of $$x^2$$ from the first two terms:** $$f(x) = -3(x^2 + 3x) + 4$$ 4. **Complete the square inside the parentheses:** - Take half the coefficient of $$x$$, which is $$3/2$$, then square it: $$\left(\frac{3}{2}\right)^2 = \frac{9}{4}$$. - Add and subtract $$\frac{9}{4}$$ inside the parentheses: $$f(x) = -3\left(x^2 + 3x + \frac{9}{4} - \frac{9}{4}\right) + 4$$ 5. **Group the perfect square trinomial and simplify:** $$f(x) = -3\left(\left(x + \frac{3}{2}\right)^2 - \frac{9}{4}\right) + 4$$ 6. **Distribute $$-3$$ and simplify constants:** $$f(x) = -3\left(x + \frac{3}{2}\right)^2 + 3 \times \frac{9}{4} + 4 = -3\left(x + \frac{3}{2}\right)^2 + \frac{27}{4} + 4$$ Convert 4 to quarters: $$4 = \frac{16}{4}$$ $$f(x) = -3\left(x + \frac{3}{2}\right)^2 + \frac{27}{4} + \frac{16}{4} = -3\left(x + \frac{3}{2}\right)^2 + \frac{43}{4}$$ 7. **Identify the vertex:** The vertex form is $$f(x) = a(x - h)^2 + k$$ with $$a = -3$$, $$h = -\frac{3}{2}$$, and $$k = \frac{43}{4}$$. So, the vertex is $$\left(-\frac{3}{2}, \frac{43}{4}\right)$$. **Final answer:** $$f(x) = -3\left(x + \frac{3}{2}\right)^2 + \frac{43}{4}$$ Vertex: $$\left(-\frac{3}{2}, \frac{43}{4}\right)$$