$\frac{ ? }{ {x}^{3} }$
Since the factor $x$ is raised to the power of $3$, it is a repeating factor, so include each power from $1$ to $3$ as denominators of separate fractions$\frac{ ? }{ x }+\frac{ ? }{ {x}^{2} }+\frac{ ? }{ {x}^{3} }$
Since the factor in the denominator is linear, the numerator is an unknown constant $A$$\frac{ A }{ x }+\frac{ ? }{ {x}^{2} }+\frac{ ? }{ {x}^{3} }$
Since the denominator is a linear factor raised to a power, the numerator is an unknown constant $B$$\frac{ A }{ x }+\frac{ B }{ {x}^{2} }+\frac{ ? }{ {x}^{3} }$
Since the denominator is a linear factor raised to a power, the numerator is an unknown constant $C$$\frac{ A }{ x }+\frac{ B }{ {x}^{2} }+\frac{ C }{ {x}^{3} }$
To get the unknown values, set the sum of fractions equal to the original fraction$\frac{ {x}^{2}+3x+2 }{ {x}^{3} }=\frac{ A }{ x }+\frac{ B }{ {x}^{2} }+\frac{ C }{ {x}^{3} }$
Multiply both sides of the equation by ${x}^{3}$${x}^{2}+3x+2=A{x}^{2}+Bx+C$
When two polynomials are equal, their corresponding coefficients must be equal$\left\{\begin{array} { l } 2=C \\ 3=B \\ 1=A\end{array} \right.$
Solve the system of equations$\left( A, B, C\right)=\left( 1, 3, 2\right)$
Substitute the given values into the formed partial-fraction decomposition$\frac{ 1 }{ x }+\frac{ 3 }{ {x}^{2} }+\frac{ 2 }{ {x}^{3} }$