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 precedes this by decades and is completely elementary, in order to verify it as follows, without the use of functional analysis (to simplify the notation, we assume that $a=0$ and $b=1$). Then it is clear that the product exists 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 manipulations is easily available online (reference below), where you will also find explicitly the following generalisations—just as easily proved:
if $f$ is a distribution on the line which equals a smooth function on a neighbourhood of $0$, then $f\delta_0$ and indeed $f\delta_0^{(n)}$ exist and are equal to zero if $f$ vanishes there.
If $a$ and $b$ are distinct, then $\delta^{(n)}(x-a)\delta^{(m)}(x-b)=0$.
The reference is to the site https://jss100.campus.ciencias.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 "Textos Didáticos".
