Solution
= Solution
Factor out $x^2$ before expanding the <logarithm>:
$$
\log(1+x+x^2)=2\log x+\log\left(1+\frac1x+\frac1{x^2}\right)
$$
for large positive $x$. Using $\log(1+u)=u-u^2/2+O(u^3)$ then gives
$$
\boxed{\log(1+x+x^2)=2\log x+\frac1x+\frac1{2x^2}+O(x^{-3}),\qquad x\to+\infty}.
$$
These are the first three nonzero terms, including the leading <logarithm>. Replacing $\log x$ by $\log|x|$ yields the corresponding two-sided large-$|x|$ expansion.