I am looking for two polynomials, $L(x)$ and $P(x)$. I know a third polynomial, $A(x) = x^4 + x^3$.
The following must be satisfied: $\deg P \leq 1 + \deg L$.
Let's define $V(x) = A(x) L(x) + P(x)$.
Question one: How do I find a candidate for $L$ and $P$ where all the roots of $V$ have negative real parts?
Question two: How do I find candidates when $A$ is left is a parameter (possibly knowing that it has multiple roots at zero)?
In general, if $\alpha_1,\dots, \alpha_{2d- 2}$ are numbers with negative real part (or any other property you want) and $d$ is the degree of $A$, set
$V = (x - \alpha_1)\dots(x - \alpha_{2d - 2})$.
The division of $V$ by $A$ has remainder $L$ of degree $d - 2$ and remainder $P$ of degree $\leq d - 1 = 1 + \deg L$ (or $P = 0$).
Note that there is no hypothesis on what $A$ looks like except for the natural assumption $A \neq 0$.