To derive the moment equation M(x) for this beam, begin by drawing a free-body diagram (FBD) that is representative of the beam in the interval 0 ≤ x ≤ L.

This FBD is almost correct. The internal moment M is shown in the wrong direction. The sign convention for a positive internal moment is clockwise on the left-hand side of a beam element. We should assume a positive direction when deriving the moment equation.