Is the set $\displaystyle \{p\in\mathbb{P}_3 : p'(x)+x+1=0 \ \textrm{for all} \ x\in\mathbb{R}\}$

a subspace of $\displaystyle \mathbb{P}_3$

Is the zero polynomial included?

I can do the closure under addition and scalar multiplication. But I can't tell if its non-empty

Let $\displaystyle p, q\in \mathbb{P}_3$

$\displaystyle (p+q)'(x)+x+1=0$

$\displaystyle \Rightarrow p'(x)+q'(x)+x+1=0$

Not closed under addition