(c) Given that ${x^2} + 2ax + {a^2}$ is a factor of

${x^3} – 3px + 2q = 0$

Let ${x^3} – 3px + 2q = ({x^2} + 2ax + {a^2})(x + \lambda )$,

where $\lambda $ is a constant.

Then equating the coefficients of like powers of x on both sides,

${x^3} – 3px + 2q = {x^3} + (2a + \lambda ){x^2} + ({a^2} + 2a\lambda )x + \lambda {a^2}$

==> $2a + \lambda = 0$$ \Rightarrow \lambda = – 2a$ …..$(i)$

and $ – 3p = {a^2} + 2a\lambda $ …..$(ii)$

and $2q = \lambda {a^2}$ …..$(iii)$

Put the value of $\lambda $ in $(iii),$

==> $2q = – 2{a^3} \Rightarrow q = – {a^3}$ ..…$(iv)$

Put the value of $\lambda $ in $(ii),$

==>$ – 3p = {a^2} + 2a( – 2a) = {a^2} – 4{a^2} = – 3{a^2}$

==> $ – 3p = – 3{a^2} \Rightarrow p = {a^2}$

==>$p = {( – q)^{2/3}}$

==> ${p^3} = {q^2}$.