You can, of course, if you wish, consult the highly sophisticated treatise of Hörmander to verify that your result holds (for distinct values of $a$ and $b$—there is no serious text which claims this for the case $a=b$). Or you can refer to work (which precede this by decades and are completely elementary) to verify it without functional analysis as follows. In oder to simplify the notation, we assume that $a=0$ and $b=1$. Then it is clear that the product exist and is equal to $0$ on each of the intervals $]-\infty,\frac 2 3[$ and $]\frac 1 3,\infty[$. We now piece these two distributions together (trivial case of “recollement des morceuses”) to get the zero distribution on the line.
We remark that the very elementary theory which justifies these simple manipulation is easily available online (where you will also find explicitly the following generalisation—just as easily proved):
if $f$ is a distribution on the line which vanishes equals a smooth function on a neighbourhood of $0$, then $f\delta_0=0$ (indeed $f\delta_0^{(n}=0$);
If $a$ and $b$ are distinct, then $\delta^{(n)}(x-a)\delta^{(m)}(x-b)=0$.
The reference is to the site
jss100.campos.ciencia.ulisboa.pt
of the late portuguese mathematician J. Sebastião e Silva where you will find his text “Theory of Distributions” (the above results are in the fourth chapter “Multiplication and Change of Variables”) under the headings “Publicações” and “Textos Didacticos”.