证明对于所有实数 $a$ 和 $b$，方程 $ax^2 + (3a + b)x + 3b = 0$ 的根都是实数。 求 $m$ 的取值范围，使得直线 $y = mx - 3$ 与曲线 $y^2 = 4x - 6y - 34$ 永不相交。

1. Write down the expression for the discriminant of the quadratic equation.

For the equation $ax^2 + (3a + b)x + 3b = 0$, the discriminant is:
$\Delta = (3a + b)^2 - 4(a)(3b)$

2. Expand and simplify the discriminant.

$\Delta = 9a^2 + 6ab + b^2 - 12ab$
$\Delta = 9a^2 - 6ab + b^2$

3. Factorise into a perfect square and conclude.

$\Delta = (3a - b)^2$
Since $(3a - b)^2 \ge 0$ for all real values of $a$ and $b$, the discriminant is always non-negative. Therefore, the roots of the equation are real for all real values of $a$ and $b$.

1. Substitute the equation of the line into the equation of the curve

Substitute $y = mx - 3$ into $y^2 = 4x - 6y - 34$:
$(mx - 3)^2 = 4x - 6(mx - 3) - 34$

2. Expand and rearrange into a standard quadratic equation in $x$

$m^2x^2 - 6mx + 9 = 4x - 6mx + 18 - 34$
$m^2x^2 - 6mx + 9 = 4x - 6mx - 16$
$m^2x^2 - 4x + 25 = 0$

3. Apply the discriminant condition for no intersection

For the line to never cut the curve, the quadratic equation must have no real roots, so $b^2 - 4ac < 0$.
Here, $a = m^2$, $b = -4$, $c = 25$.
$(-4)^2 - 4(m^2)(25) < 0$
$16 - 100m^2 < 0$

4. Solve the quadratic inequality for $m$

$100m^2 > 16$
$m^2 > \frac{16}{100} = \frac{4}{25}$
$m < -\frac{2}{5} \text{ or } m > \frac{2}{5}$