= Solution
Factor the <Gaussian function> as $e^{-(x-a)^2}=e^{-a^2}e^{2ax}e^{-x^2}$. Multiplying the two <exponential series> gives
$$
e^{-(x-a)^2}=e^{-a^2}\left[1+2ax+(2a^2-1)x^2+\left(\frac43a^3-2a\right)x^3+O(x^4)\right].
$$
For $a\ne0,\pm1/\sqrt2$, its first three nonzero terms are
$$
\boxed{e^{-a^2}\bigl[1+2ax+(2a^2-1)x^2\bigr].}
$$
Vanishing coefficients require separate treatment in this <Taylor expansion of a translated Gaussian>. If $a=0$, the first three nonzero terms are $\boxed{1-x^2+x^4/2}$. If $a=\pm1/\sqrt2$, they are
$$
\boxed{e^{-1/2}\left(1+2ax-\frac{4a}3x^3\right).}
$$
The latter expression retains the cubic term because the quadratic coefficient is zero.
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