Does the equality $\omega\cdot(\omega+1)=(\omega+1)\cdot\omega$ hold?

Question

By this answer I knkow that the equality $$ (\omega+1)\cdot\omega=\omega^3 $$ holds whereas by the definition of ordinal multiplication I know that the equality $$ \omega\cdot(\omega+1)=\omega^2+\omega $$ holds: so it seems to me that the inequality $$ \omega^2+\omega<\omega^3 $$ holds so that the equality $$ \omega\cdot(\omega+1)=(\omega+1)\cdot\omega $$ does not hold but I was not able to prove or disprove this. So could someone prove or disprove if $\omega\cdot(\omega+1)$ and $(\omega+1)\cdot\omega$ are equal or not, please?