Express $-4x^2 + 12x + 2$ in the form $a(x + b)^2 + c$ and state the coordinates of the turning point of the curve $y = -4x^2 + 12x + 2$. Hence explain why the turning point is a maximum point.

1. Factor out the coefficient of $x^2$ from the $x$ terms

$-4x^2 + 12x + 2 = -4(x^2 - 3x) + 2$

2. Complete the square inside the bracket

$-4\left[ \left(x - \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 \right] + 2$
$= -4\left[ \left(x - \frac{3}{2}\right)^2 - \frac{9}{4} \right] + 2$

3. Expand and simplify to reach the final form

$= -4\left(x - \frac{3}{2}\right)^2 + 9 + 2$
$= -4\left(x - \frac{3}{2}\right)^2 + 11$

4. State the coordinates of the turning point

The turning point is at $\left(\frac{3}{2}, 11\right)$.

1. Use the completed square form from part (a) to explain the maximum value

From part (a), $y = -4(x - 1.5)^2 + 11$.
For any real value of $x$, $(x - 1.5)^2 \ge 0$.
Therefore, $-4(x - 1.5)^2 \le 0$.

2. Conclude that the turning point is a maximum

Since the term $-4(x - 1.5)^2$ is always negative or zero, the highest possible value for $y$ is $11$ (when $x = 1.5$). Hence, the turning point is a maximum point.