Solutions $3 p\sin x - (p+\sin x)(p^2-p \sin x +\sin ^{2} x) =1$

Question

$3 p \sin x - (p+\sin x)(p^2-p \sin x + \sin ^{2} x) =1$ has a solution for $x$. Then number of integral solutions of $p$ are ?

Answer 1

On simplifying the the expression is $3p \sin x - p^{3} -\sin^{3}x=1$ which can be written as $(p)^{3}+(\sin x)^3+(1)^3=3 \cdot 1 \cdot p \cdot \sin x$, i.e $a^3+b^3+c^3=3abc$ form. This means either $p+\sin x+1=0$ or $p=\sin x =1$ Hence possible values of p are $-2,-1,0,1$, i.e 4 integer solutions.