My teacher insists that synthetic division could be done with 4x-5, by setting the divisor to -5/4 and reworking the quotient to get a correct answer. Doesn't that defeat the purpose really?

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Why in the world would you use synthetic division for that? It's not even a polynomial function. Synthetic division just makes factoring easier when you have a term that's higher than x^2....like x^3+6x^2+9x+16 (just an example, let's just say you can do synthetic division with it). Don't worry about it....x_x

what they said below