A curve has equation $y = x^2 + x + 2k - 4$ and a line has equation $y - 2x = k$, where $k$ is a constant. Find the set of values of $k$ for which the line and curve do not intersect and represent this set on a number line. Given that the curve with equation $y = ax^2 + 8x + c$, where $a$ and $c$ are constants, lies completely below the $x$-axis.
Write down the conditions which must apply to $a$ and $c$. If $x$ is increasing at a rate of $0.2$ units per second, find the rate of change of $y$ at the instant when $y = -4$.

1. Express $y$ in terms of $x$ from the linear equation and substitute into the curve equation

From the line equation, $y = 2x + k$.
Substitute into the curve equation:
$2x + k = x^2 + x + 2k - 4$

2. Rearrange into a standard quadratic equation form $ax^2 + bx + c = 0$

$x^2 + x - 2x + 2k - k - 4 = 0$
$x^2 - x + k - 4 = 0$

3. Apply the discriminant condition for no intersection

For the line and curve to not intersect, the quadratic equation must have no real roots, so $b^2 - 4ac < 0$.
$(-1)^2 - 4(1)(k - 4) < 0$

4. Solve the inequality for $k$

$1 - 4k + 16 < 0$
$17 - 4k < 0$
$4k > 17$
$k > 4.25$

5. Represent the solution on a number line

Draw a number line, place an open circle at $4.25$ (or $\frac{17}{4}$), and draw an arrow pointing to the right to indicate values greater than $4.25$.

1. State the condition on the leading coefficient for the curve to lie completely below the $x$-axis

For the curve to lie completely below the $x$-axis, it must open downwards. Therefore, the coefficient of $x^2$ must be negative:
$a < 0$

2. Apply the discriminant condition for the curve not to intersect the $x$-axis

The curve must not intersect the $x$-axis, so the equation $ax^2 + 8x + c = 0$ has no real roots. Thus, the discriminant must be less than zero:
$b^2 - 4ac < 0$
$(8)^2 - 4(a)(c) < 0$
$64 - 4ac < 0$
$4ac > 64$
$ac > 16$