show that the equation $x^5+5px^3+5p^2x+q=0$ will have a pair of equal roots, if $q^2+4p^5=0$ how can I show that the equation $x^5+5px^3+5p^2x+q=0$ will have a pair of equal roots, if $q^2+4p^5=0$.
can anyone help me.thanks a lot.

A polynomial has a pair of equal roots if it has a common factor with its derivative, so you might want to compute the derivative and then see whether that condition gives it a common root with the original polynomial. 
There is a more systematic approach, involving the resultant (which you can look up). The polynomial and its derivative have a common root if and only if the resultant of the polynomial and its derivative is zero. So, you can calculate the resultant, then see whether $q^2+4p^5$ is a factor of the resultant. 
These seem like unlikely approaches for something at the level of algebara-precalculus, though. 
