SOLUTION: Find the relationship between Q and R so that X³+3PX²+QX+R shall be a perfect cube for all values of X

Let S be such that

X³+3PX²+QX+R = (X+S)³

X³+3PX²+QX+R = X³+3X²S+3XS²+S³

Each term in a power of X on the left must equal 
identically to the corresponding power of X on the 
right side, and also the corrsponding constant terms 
must also be equal:

X³ = X³,  3PX² = 3X²S,  QX = 3XS²,  R = S³
            3P = 3S      Q = S²     
             P = S

Multiply equals by equals using P = S and Q = S²

PQ = S³

And since also R = S³,

PQ = R   <-- answer.

Edwin