I'm trying to prove that the divergence of the energy-momentum-tensor is zero by expressing it in terms of the field strength tensor: $\partial_\mu T^{\mu\nu}=0$.

In doing this, letting the derivative act on the second part of the tensor (the Lagrangian part) one obtains the following term (as a part of the full term) after applying the product rule: $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}+F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$.
Here the solution asserts that this is equal to simply equal to twice the first part of the term, implying $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$.
Can anyone help me in understanding why this is the case? I have not been able to find many resources on this and those (or asking AI lol) have not proven enlightning to me (so far).