There is an image link in there which is neither showing nor working
edit tis now, give me some time for working please
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1. Fine up until the penultimate step, however on the last line there are algebraic errors, you cannot cancel $\displaystyle x^2$ and $\displaystyle (x+0.5)^2 \neq x^2 + 0.5^2$
From your penultimate step: $\displaystyle g'(x) = \dfrac{x^2+x+4}{(x+0.5)^2}$
I'd leave it in that form