Substitute $z(x)={\displaystyle \underset{n=0}{\overset{}{}}}{a}_n{x}^n$ into $z^{\text{'}\text{'}}+x{z}^{\text{'}}+z=x^2+3x+5$ and equate the coefficients of like powers of $x$ on both sides of the equation to find the first four nonzero terms in a power series expansion about $x=0$ of a general solution to the differential equation.

$z(x)=$

$+$cdots

$($Type an expression in terms of $a_0$ and $a_1$ that includes all terms up to order $3.)$