When i solve this partial fraction, i end up with Imaginary terms
how he has done it
kindly explain this to me
It is not difficult to learn some basic latex, sufficient to write fractions and express indices. Click on the link in my signature. If you want people to make an effort and give help, you need to make an effort too.
Since the factor $\displaystyle s^2 + s + 10$ in the denominator of the expression is a non-repeated irreducible quadratic, the term that factor contributes to the partial fraction form is $\displaystyle \displaystyle{ \frac{as + b}{s^2 + s + 10}}$.
Knowing the original question might shed some light on what you're trying to do and why.
You were asked to show all the work you did. You said in your first post that you got imaginary values but in
"all" the work you show you have not even attempted to solve for A, B, and C.
If you insist upon writing the fractions with linear, complex, denominators, then, yes, you will get imaginary values for B and C. But normally, we don't want that- we want real denominators and real coefficients. Instead try writing the fraction as
$\displaystyle \frac{10(s+1)}{s(s^2+ s+ 10)}= \frac{A}{s}+ \frac{Bs+ C}{s^2+ s+ 10}$
as mr fantastic suggests. Putting in three values for s will give 3 equations for real values of A, B, and C.