Solving equation with complex roots

I'm solving the equation $$w^2 = \frac{-11}{4} -15i$$
and I should thereafter use the answer to solve
$$z^2-(3+4i)z+(1+21i)=0$$

I solve the first equation and get answers
$$w_1 = \frac{5}{2}-3i$$ and $$w_2=-\frac{5}{2}+3i$$

I know how to go about part two of the question,
but I can't figure out when I should use answers
from part one of the question
to solve part two.

What I've so far tried is
that I've completed the brackets in part two:

$$(z-3-4i)^2-(-3-4i)^2+(1+21i)=0$$

I've then substituted
$(z-3-4i)^2$ for $w^2$
and set $w=(w_1)$
and in another part $w=(w_2)$
and then solved.

I get answers
$$z_1=-\frac{71}{4}-8i$$

and
$$z_2=-\frac{107}{4}-18i$$
which, when controlled,
doesn't equate the original equation to zero.

Any help or suggestions is highly appreciated!