Mistake or not - Fourier Series of x(2t+3)
Ok, I see now: They did it straight from the computation. The 3/2 is because they divided T by two in $e^{j3k{\frac {2\pi} T}}$.
It’s important, because it tells us that the first intuition we have that we can apply scaling to Fourier Series or Fourier Transform and then apply transformation to the Fourier Transform  is wrong. We might mistakenly think that we can scale time, apply transform, and then apply shift by 3 to the transform to move by 3 units in time. As you can see it does not produce the same result, since the shift is by 3/scaling-factor and not straight 3.