First, I am suppose to find a prime $p\geq 4$ where $x+10$ divides $x^4+x^3+x+1$ in $\mathbb{Z}_p[x]$. Second, I am supposed to find a fifth degree polynomial in $\mathbb{Z}_2[x]$ that is reducible but has no roots in $\mathbb{Z}_2$.
For the first one, I thought $p=11$ would work okay. For the second, what about $x^5+x+1=(x^3+x^2+1)(x^2+x+1)$? Do these examples work okay? Or any nicer/cleaner ones?
$\mathbb{F}_p[x]$ is an integral domain, hence $x+10$ is a divisor of $$q(x)=x^4+x^3+x+1=(x^2-x+1)(x+1)^2$$ iff $q(-10)\equiv 0\pmod{p}$. Since $q(-10)=8991=3^5\cdot 37$, $\color{red}{p=37}$ is the only chance.
For the second point, we just need to find two irreducible polynomials over $\mathbb{F}_2$ having degree $2$ and $3$. Since: $$ q_2(x) = x^2+x+1,\qquad q_3(x)=x^3+x^2+1 $$ are both trinomials, they have no root in $\mathbb{F}_2$, so they are irreducible over $\mathbb{F}_2$ and your $$ x^5+x+1 = q_2(x)\cdot q_3(x) $$ solves the problem.