If $$|z + 4| \le 3$$, then the maximum value of $$|z + 1|$$ is

The condition $$|z+4|\le 3$$ represents the closed disc in the complex plane whose centre is the point $$-4$$ (on the real axis) and whose radius is $$3$$.

We must find the greatest possible value of $$|z+1|$$, i.e. the farthest distance from the fixed point $$-1$$ to any point $$z$$ lying in (or on) that disc.

Let us denote the centre of the disc by $$C=-4$$ and the reference point by $$P=-1$$.

Distance between the two fixed points is
$$CP = |-1-(-4)| = |-1+4| = 3.$$

For any circle (or disc) of radius $$r$$ centred at $$C$$, the maximum distance from an external (or internal) point $$P$$ to a variable point $$Z$$ on the circle is given by
$$\text{(distance }CP) + r.$$

Here $$r = 3$$ and $$CP = 3$$, hence
$$\max |z+1| = CP + r = 3 + 3 = 6.$$

Therefore, the required maximum value is $$6$$.

Option C which is: 6