Find the range of values of the constant $m$ for which the curve $y = (m - 3)x^2 + mx + m$ lies completely above the $x$-axis.

1. State the conditions for the quadratic curve to lie completely above the x-axis

For the curve $y = (m - 3)x^2 + mx + m$ to lie completely above the $x$-axis, it must be a U-shaped parabola that does not intersect the $x$-axis.
Conditions:

1. Coefficient of $x^2 > 0 \implies m - 3 > 0$
2. Discriminant $< 0 \implies b^2 - 4ac < 0$

2. Solve the condition for the coefficient of x^2

$m - 3 > 0 \implies m > 3$

3. Set up the discriminant inequality

$b^2 - 4ac < 0$
$(m)^2 - 4(m - 3)(m) < 0$

4. Solve the discriminant inequality

$m^2 - 4m^2 + 12m < 0$
$-3m^2 + 12m < 0$
$-3m(m - 4) < 0$
$m(m - 4) > 0$
$m < 0$ or $m > 4$

5. Combine the conditions to find the final range of m

Intersection of $m > 3$ and ($m < 0$ or $m > 4$):
$m > 4$