Possible value of $( p + q )$. Let $ g : \Bbb R \to ( - \infty, - 1]$ be the function defined as $$g(x) = ( pq + 2p - q - 2 ) x^5 -( p^3 - 2p + 1 ) x^3 + ( p^2 - 2p - 3 ) x^2 + ( p^2 + 2q )x - 5$$ where $p,q \in \Bbb R$. If $g(x)$ is surjective, then the possible value of $( p + q )$ is (are)

Hint...

The range is restricted. Hence the polynomial can't be an odd degree polynomial. So make the coefficients of $x$ having odd powers equal to $0$ to find value of p.

Note - Don't make coefficient of $x$ as $0$ . Just set coefficients of $x^5$ and $x^3$ equal to zero, so that the polynomial immediately becomes even degreed.

Now you know the maximum value of quadratic so obtained. Hence find the maximum value in terms of $q$ and set it equal to $-1$ to get possible values of $q$