Express $5 - 4x - 2x^{2}$ in the form $c - a (x + b)^{2}$, where $a$, $b$ and $c$ are constants and $a > 0$. Sketch the graph of $y = 5 - 4x - 2x^{2}$, indicating clearly the turning point and the y-intercept. Hence find the range of values of $k$ for which the equation $5 - 4x - 2x^{2} = k$ has at most one root.

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

$5 - 4x - 2x^2 = 5 - 2(x^2 + 2x)$

2. Complete the square inside the bracket

$5 - 2\left[ (x + 1)^2 - 1^2 \right]$

3. Expand and simplify to the required form

$5 - 2(x + 1)^2 + 2 = 7 - 2(x + 1)^2$

1. Identify the turning point using the completed square form from part (i)

From part (i), $y = 7 - 2(x+1)^2$.
The maximum turning point occurs when $x + 1 = 0 \implies x = -1$.
The maximum value is $y = 7$.
Turning point is $(-1, 7)$.

2. Identify the y-intercept and sketch the graph

When $x = 0$, $y = 5 - 4(0) - 2(0)^2 = 5$.
The y-intercept is $(0, 5)$.
Since the coefficient of $x^2$ is $-2 < 0$, the parabola opens downwards (inverted U-shape).
Sketch the curve passing through $(0, 5)$ with a peak at $(-1, 7)$.

1. Relate the number of roots to the maximum value of the quadratic function

From part (i), $5 - 4x - 2x^{2} = 7 - 2(x+1)^{2}$.
The maximum value of the curve $y = 5 - 4x - 2x^{2}$ is $7$ (at the turning point).
The equation $5 - 4x - 2x^{2} = k$ corresponds to the intersection of the curve and the horizontal line $y = k$.

2. Determine the range of $k$ for at most one root

For the equation to have at most one root, the line $y = k$ must intersect the curve at exactly one point (tangent at the maximum) or not intersect it at all (above the maximum).
Therefore, $k \geqslant 7$.