<s>[INST] Polynomial equation Is it possible to find polynomials with rational coefficients $P(x),Q(x)$ such that $Px^3+Px^2+Qx+2Q=1$? I have trying in vain to find one by inspection, but that might just be me.
 [/INST] Just in case the OP does not know the Euclidean algorithm, I'll compute it.
Divide $(x^3 + x^2)$ by $x+2$. This is just ordinary high school long division. So we find that
$$\begin{equation}(x^3 + x^2) = (x+2)q(x) + r(x)\end{equation}$$
where $q(x) = x^2 - x$ and $r(x) = 2x$. Now by the Euclidean algorithm we then divide $(x+ 2)$ by $2x$ to get
$(x+2) = 2x(\frac{1}{2}) + 2$ so that $2 = (x+2) - 2x(\frac{1}{2})$. Now from the first expression we found that $2x = (x^3 + x^2) - (x+2)(x^2 - x)$, so that substituting it in here we get
$$\begin{eqnarray*}2 &=& (x+2) - \frac{\bigg[(x^3 + x^2) - (x+2)(x^2 - x)         \bigg]}{2}\\
&=& (x+2)\frac{(2+ (x^2 - x))}{4} - \frac{[x^3 + x^2]}{4} \end{eqnarray*} $$
Reading off this your $P$ is $-1/4$ while $$ Q= \frac{(2+ (x^2 - x))}{4} .$$
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