How do you divide the complex number \[\dfrac{4-9i}{-6i}\]?

Hint: In this problem, we have to divide the given complex fraction. We know that we can take the complex conjugate of the denominator and we can multiply the resulting complex conjugate to both the numerator and the denominator to get a simplified form. We will also use some complex formula to get some real numbers.

Complete step-by-step solution:
We know that the given fraction is,
\[\dfrac{4-9i}{-6i}\]
We can now find the complex conjugate of the denominator.
the conjugate of the denominator \[-6i\] is \[6i\].
We can now multiply the complex conjugate in both the numerator and the denominator, we get
\[\Rightarrow \dfrac{4-9i}{-6i}\times \dfrac{\left( 6i \right)}{\left( 6i \right)}\]
We can now multiply every term in both the numerator and the denominator, we get
\[\Rightarrow \dfrac{24i-54{{i}^{2}}}{-36{{i}^{2}}}\]
We also know that in complex numbers,
\[{{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]
We can substitute the above value in the above step, we get
\[\Rightarrow \dfrac{24i-54\left( -1 \right)}{-36\left( -1 \right)}\]
Now we can simplify the above step, we get
\[\Rightarrow \dfrac{24i+54}{36}\]
We can see that we have two terms in the numerator with one denominator, we can separate it, we get
\[\Rightarrow \dfrac{54}{36}+\dfrac{24}{36}i\]
We can now further simplify the above step by cancelling the terms using multiplication tables.
\[\Rightarrow \dfrac{3}{2}+\dfrac{2}{3}i\]
Therefore, the answer is \[\dfrac{3}{2}+\dfrac{2}{3}i\].

Note: Students should remember the basic complex formulas to be used in these types of problems such as the values for \[{{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1\]. We should also concentrate on the part where we find the complex conjugate by changing the imaginary part. We should also know to provide the answer to its simplest form.