Express $y = -2x^2 - 12x + 1$ in the form $y = a - 2(x + b)^2$ and hence state the maximum value of $y$, and its corresponding value of $x$.

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

$y = -2(x^2 + 6x) + 1$

2. Complete the square inside the bracket

$y = -2\left[\left(x + \frac{6}{2}\right)^2 - \left(\frac{6}{2}\right)^2\right] + 1$
$y = -2[(x + 3)^2 - 9] + 1$

3. Expand and simplify to the required form

$y = -2(x + 3)^2 + 18 + 1$
$y = 19 - 2(x + 3)^2$

4. State the maximum value and corresponding $x$

Maximum value of $y$ is $19$.
This occurs when $x + 3 = 0 \implies x = -3$.