For solving integration-related questions, a rational proper fraction of the form $\frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)}$ is decomposed into the sum of the expressions, $$\frac{A}{x-a} + \frac{Bx+C}{x^{2}+bx+c}$$ where $x^{2}+bx+c$ can't be factorised further.
I don't understand why we have to use $Bx+C$ in the numerator of the second expression. Why can't it be just $B$?
To elaborate, if $\frac{x^{2}+x+1}{(x-4)(x^{2}+x+3)}$ is decomposed into the sum $\frac{A}{x-4} + \frac{B}{x^{2}+x+3}$, we will get contradictory values for $A$ and $B$. But if we use $Bx+C$ in the second expression, the method works fine. Why is this so?