[INST] By partial fractions,

 1/x(x + 1)(x + 2)(x + 3)(x + 4) = A/x + B/(x + 1) + C/(x + 2) + D/(x + 3) + E/(x + 4)
for some constants A, B, C, D, and E. Find A + B + C + D + E./n [/INST]  ###Response: We want to express 1/x(x + 1)(x + 2)(x + 3)(x + 4) in the form A/x + B/(x + 1) + C/(x + 2) + D/(x + 3) + E/(x + 4), where A, B, C, D, and E are constants. 

The partial fraction decomposition gives us:

1/x(x + 1)(x + 2)(x + 3)(x + 4) = A/x + B/(x + 1) + C/(x + 2) + D/(x + 3) + E/(x + 4)

To find the value of A + B + C + D + E, we can compare coefficients of x^4 on both sides. However, it's easier to notice that both sides represent the same polynomial, so the polynomial on the right must simplify to 1. Thus, we have:

A(x + 1)(x + 2)(x + 3)(x + 4) 
+ Bx(x + 2)(x + 3)(x + 4) 
+ Cx(x + 1)(x + 3)(x + 4) 
+ Dx(x + 1)(x + 2)(x + 4) 
+ Ex(x + 1)(x + 2)(x + 3) = 1

Since the coefficient of x^4 on the right-hand side is 0, we have:

A + B + C + D + E = 0.