(-3+6i):(5-2i)

To simplify the expression \((-3 + 6i) : (5 - 2i)\), you can rewrite this as a division of two complex numbers:

\[ \frac{-3 + 6i}{5 - 2i} \]

To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5 - 2i\) is \(5 + 2i\). Thus:

\[ \frac{-3 + 6i}{5 - 2i} \cdot \frac{5 + 2i}{5 + 2i} = \frac{(-3 + 6i)(5 + 2i)}{(5 - 2i)(5 + 2i)} \]

Calculate the denominator first:

\[ (5 - 2i)(5 + 2i) = 5^2 - (2i)^2 = 25 - (-4) = 25 + 4 = 29 \]

Now calculate the numerator:

\[ (-3 + 6i)(5 + 2i) = -3 \cdot 5 + (-3) \cdot (2i) + (6i) \cdot 5 + (6i)(2i) \]

Calculating each term:

\[ -15 - 6i + 30i + 12i^2 \]

Since \(i^2 = -1\), we have:

\[ -15 - 6i + 30i + 12(-1) = -15 + 12 + 24i = -3 + 24i \]

Putting it all together, we get:

\[ \frac{-3 + 24i}{29} \]

This can be written as:

\[ -\frac{3}{29} + \frac{24}{29}i \]

So the simplified form of \((-3 + 6i) : (5 - 2i)\) is:

\[ -\frac{3}{29} + \frac{24}{29}i \]