Calculus of a Single Variable, Chapter 2, 2.3, Section 2.3, Problem 28

You need to find derivative of the function using the product rule:
f'(x)= (x^4)'*((x+1-2)/(x+1)) + (x^4)*((x-1)/(x+1))'
You need to use the quotient rule to differentiate ((x-1)/(x+1)):
f'(x)= (4x^3)*((x-1)/(x+1)) + (x^4)*((x-1)'*(x+1) - (x-1)*(x+1)')/((x+1)^2)
f'(x)= (4x^3)*((x-1)/(x+1))+ (x^4)*(x+1 - x + 1)/((x+1)^2)
Reducing like terms yields:
f'(x)= (4x^3)*((x-1)/(x+1))+ 2*(x^4)/((x+1)^2)
You need to factor out (2x^3)/(x+1):
f'(x)= (2x^3)/(x+1)*(2(x-1)+ x/(x+1))
f'(x)= (2x^3)/(x+1)*((2x^2 + x - 2)/(x+1))
Hence, evaluating the derivative of the function, yields f'(x)= (2x^3)/(x+1)*((2x^2 + x - 2)/(x+1)).