Answer
The partial fraction expansion is $\frac{5{{x}^{2}}-9x+19}{\left( x-4 \right)\left( {{x}^{2}}+5 \right)}=\frac{A}{x-4}+\frac{Bx+C}{{{x}^{2}}+5}$.
Work Step by Step
The provided rational expression is as given below:
$\frac{5{{x}^{2}}-9x+19}{\left( x-4 \right)\left( {{x}^{2}}+5 \right)}$
Now, solve the expression as follows:
We set up the partial fraction expansion with unknown constants coefficients and then write a constant coefficients over each of the two distinct algebraic linear factors in the denominator of the expression.
Then, decompose the fractional part as follows:
$\frac{5{{x}^{2}}-9x+19}{\left( x-5 \right)\left( {{x}^{2}}+5 \right)}=\frac{A}{x-4}+\frac{Bx+C}{{{x}^{2}}+5}$
Thus, $\frac{A}{x-4}+\frac{Bx+C}{{{x}^{2}}+5}$ is a partial fraction expansion of the rational expression $\frac{5{{x}^{2}}-9x+19}{\left( x-4 \right)\left( {{x}^{2}}+5 \right)}$with constants $A$,$B$ and$C$.
