This is very peculiar- you use the terms "Taylor Polynomial" and "Taylor Series" but your questions imply that you don't know the definition or formula!
The "third degree Taylor Polynomial" for function f(x) about x= a is
\(\displaystyle \frac{f'''(a)}{3!}(x- a)^3+ \frac{f''(a)}{2!}(x- a)^2+ \frac{f'(a)}{1}x+ f(a)\).
Here, \(\displaystyle f(x)= (7x+ 228)^{5/4}\). \(\displaystyle f(4)= (28+ 228)^{5/4}= 4^5= 1024\).
\(\displaystyle f'(x)= \frac{5}{4}(7x+ 228)^{1/4}(7)= \frac{35}{4}(7x+ 228)^{1/4}\) so \(\displaystyle f'(4)= \frac{35}{4}(28+ 228)^{1/4}= 35\).
Can you get f''(4) and f'''(4)?