Question: User: g(x) = -2(x^2 + 3x + 9/4 - 9/4) + 9
Remove -9/4 from within parentheses along with a factor of -2.
g(x) = -2(x^2 + 3x + 9/4) - 2(-9/4) + 9
g(x) = -2(\square)^2 + \square
graph shape: parabola opening downward, with the squared term centered at the top-left/center area and two blanks for vertex-form completion in the bottom-center; position_hint=center
1. **State the problem:**
Rewrite the function $$g(x) = -2\left(x^2 + 3x + \frac{9}{4} - \frac{9}{4}\right) + 9$$ by removing the term $$-\frac{9}{4}$$ inside the parentheses and adjusting the expression accordingly.
2. **Rewrite the expression by distributing the factor $$-2$$:**
$$g(x) = -2\left(x^2 + 3x + \frac{9}{4}\right) - 2\left(-\frac{9}{4}\right) + 9$$
3. **Simplify the term outside the parentheses:**
Calculate $$-2\left(-\frac{9}{4}\right) = \frac{18}{4} = \frac{9}{2}$$.
4. **Rewrite the function:**
$$g(x) = -2\left(x^2 + 3x + \frac{9}{4}\right) + \frac{9}{2} + 9$$
5. **Simplify the constants:**
$$\frac{9}{2} + 9 = \frac{9}{2} + \frac{18}{2} = \frac{27}{2}$$
6. **Recognize the perfect square trinomial inside the parentheses:**
$$x^2 + 3x + \frac{9}{4} = \left(x + \frac{3}{2}\right)^2$$
7. **Write the function in vertex form:**
$$g(x) = -2\left(x + \frac{3}{2}\right)^2 + \frac{27}{2}$$
**Final answer:**
$$g(x) = -2\left(x + \frac{3}{2}\right)^2 + \frac{27}{2}$$
This matches the form $$g(x) = -2(\square)^2 + \square$$ where the squared term is $$\left(x + \frac{3}{2}\right)^2$$ and the constant term is $$\frac{27}{2}$$.