____________  
(-3i2 + 4i - 5)Ö-9 - 4 + 5i3

Combine the -9 and the -4 under the radical

                _________  
(-3i2 + 4i - 5)Ö-13 + 5i3


Since  i2 = -1 and i3 = -i, replace those
                    ___________
( -3(-1) + 4i - 5 )Ö-13 + 5(-i)
              ________
( 3 + 4i - 5)Ö-13 - 5i
            ________  
( -2 + 4i )Ö-13 - 5i

Assume this equals to the complex number A + Bi, where
A and B are real numbers

          ________  
(-2 + 4i)Ö-13 - 5i = A + Bi

Square both sides:

            ________  
(-2 + 4i)2(Ö-13 - 5i)2 = (A + Bi)2

(4 - 16i + 16i2)(-13 - 5i) = A2 + 2ABi + B2i2

Replace the i2's by -1

( 4 - 16i + 16(-1) )(-13 - 5i) = A2 + 2ABi + B2(-1)

( 4 - 16i - 16 )(-13 - 5i) = A2 + 2ABi - B2

(-12 - 16i)(-13 - 5i) = A² + 2ABi - B2

156 + 60i + 208i + 80i2 = A2 + 2ABi - B2

Replace the i2 by -1

156 + 60i + 208i + 80(-1) = A2 + 2ABi - B2

156 + 60i + 208i - 80 = A2 + 2ABi - B2

76 + 268i = A2 + 2ABi - B2

The real numbers on the left must equal the real
numbers on the right, so

76 = A2 - B2

A2 - B2 = 76

The imaginary numbers on the left must equal the
imaginary numbers on the right, so

268i = 2ABi

Dividing thru by 2i

134 = AB

 AB = 134

So we have this system of equations:

A2 - B2 = 76
     AB = 134

To make things easier square the second equation

    A2B2 = 17956

Solve it for B2

      B2 = 17956/A2

Substitute for B2 in the first equation

A2 - 17956/A2 = 76

A4 - 17956 = 76A2

A4 - 76A2 - 17956 = 0

Solve that for A2 by the quadratic formula

A2 = 177.2838828, A2 = -101.2838828

But since A is real A2 is positive so we discard
the negative value

A2 = 177.2838828

Taking square roots:

A = ±13.31479939

Substitute 177.2839928 for A2 in

A2 - B2 = 76

177.2838828 - B2 = 76

177.2838828 - 76 = B2

101.2838828 = B2

B2 = 101.2838828

Taking square roots:

B = ± 10.06398941

So we get four solutions:

A + Bi =  13.31479939 + 10.06398941i
A + Bi =  13.31479939 - 10.06398941i
A + Bi = -13.31479939 + 10.06398941i
A + Bi = -13.31479939 - 10.06398941i

However we can eliminate two of these because
of the equation

     AB = 134

since 134 is a positive number, which means
that A and B must have the same sign.  This
only leaves the two solutions:

A + Bi =  13.31479939 + 10.06398941i
A + Bi = -13.31479939 - 10.06398941i

Both those are correct answers because
there are two complex imaginary square
roots of a complex imaginary number.
The original problem contains
a square root of a complex number, thus
we expect two answers.

Edwin