nsider the following quadratic function.
\[
f(x)=-3 x^{2}+6 x+4
\]
(a) Write the equation in the form $f(x)=a(x-h)^{2}+k$.
Writing in the form specified: $f(x)=$

Step 1 ：The given equation is in the standard form of a quadratic equation, which is \(f(x) = ax^2 + bx + c\). We need to convert it into the vertex form, which is \(f(x) = a(x-h)^2 + k\). The vertex form is useful because it clearly shows the vertex of the parabola, which is \((h, k)\).

Step 2 ：To convert the standard form to the vertex form, we can complete the square. The steps to complete the square are as follows: 1. Factor out the coefficient of \(x^2\) from the first two terms on the right side of the equation. 2. Inside the parentheses, add and subtract the square of half the coefficient of \(x\). 3. Simplify the equation by combining like terms.

Step 3 ：Let's apply these steps to the given equation.

Step 4 ：The equation in the form \(f(x) = a(x-h)^2 + k\) is \(f(x) = 7 - 3(x - 1)^2\). We can see that the expanded form of this equation is the same as the original equation, which confirms that the conversion is correct.

Step 5 ：Final Answer: The equation in the form \(f(x) = a(x-h)^2 + k\) is \(\boxed{7 - 3(x - 1)^2}\).