Can someone explain to me why is the answer $30b^{3}/4(a+b)$ considering that on the previous line we multiply $[5b][a+b]/{[4][6b^2]}$. It's as if we multiply the numerator of the first term with the denominator of the second term. Is there a special rule that I'm missing?
The last line in the process you have described is just wrong. You seem to correctly understand that the result is $\frac{5ab+5b^2}{24b^2}$.