Consider the equation $x^4+a_0x^3+a_1x^2+2a_2x+a_3=0$. Prove that there is $\delta > 0$ such that if $|a_i-1| < \delta$ for $i=0,1,2,3$, then the above equation has a solution that depends smoothly on the $a_i$'s.
I assume this is an exercise on the Implicit function theorem, but I don't know how to apply it in this case. I thought about the function $\mathbb R^4\to \mathbb R, (a_0,a_1,a_2,a_3)\mapsto x^4+a_0x^3+a_1x^2+2a_2x+a_3$, but the theorem requires that the function vanishes at some point, whereas this function contains a formal variable and cannot vanish.
The function you have defined is not a map to $\mathbb{R}$. It is a map to $\mathbb{R}[x]$. Almost certainly you want a map $\mathbb{R}^5 \to \mathbb{R}$ which is defined by $(a_0, a_1, a_2, a_3, x) \mapsto x^4 + a_0x^3 + a_1x^2 + 2a_2x + a_3 $. In this case, if you think of this as a function $\mathbb{R}^{4+1} \to \mathbb{R}^1$, you'll find the implicit function theorem to be a little more friendly. In particular, if you take for example all the $a_i = 1$, you can find a solution given by $x = -1$. Now you just need to show that $x$ can be eliminated in terms of the $a_i$ by using the implicit function theorem, i.e. checking that the $x$ derivative of this function of $5$ variables has a non-zero derivative at the point in question (as it is clearly more than just $C^1$, it is $C^\infty$ as a polynomial in these variables).