What conditions must the coefficients $a, b, c, d$ satisfy so that the function

$$
f(x)=x^{4}+a x^{3}+b x^{2}+c x+d
$$

can be written in the form

$$
\varphi(z)=z^{4}+A z^{2}+B
$$

?
Let $f(x)$ be denoted by $y$, and $\varphi(z)$ by $v$. The image of function (1) is a curve (a quartic parabola) in the coordinate system whose axes are $x$ and $y$. Our task is to derive the equation of this curve in the form (2) in a new coordinate system with axes $z$ and $v$, and to specify the condition for the transformability into form (2). (We note that we are only looking for a linear, i.e., first-degree, transformation.)

Our goal can only be met by a transformation that maps the lines

$$
x=k \text { (constant) }
$$

onto the lines

$$
z=k^{*} \text { (constant) }
$$

Indeed, $y=f(x)$ is a single-valued function of $x$ (for any value of $x$, there is one and only one value of $y$), and similarly, $v=\varphi(z)$ is a single-valued function of $z$. However, if the line $z=k^{*}$ corresponds in the original $(x, y)$ coordinate system to a line not of the form $x=k$, but rather of the form

$$
y=m x+k
$$

then after the transformation, the four intersection points of (1) and (5) would correspond to four function values for $z=k^{*}$, and thus the transformed function could not be a single-valued function of $v$ in the form (2).

The transformation that meets our goal has the new ordinate axis $v$ parallel to $y$, so the following relationship holds between $x$ and $z$

$$
x=z+\lambda
$$

where $\lambda$ is a constant. The new system's abscissa axis can coincide with the $x$ axis, so

$$
v=y
$$

because a parallel shift of the abscissa axis (2) only changes the constant term, which does not affect the form (2).

(Apart from the simple parallel shift, a transformation could also be considered where we first translate the coordinate system and then rotate it by $180^{\circ}$. The rotation changes the sign of both coordinates, so it transforms (1) into a quartic function whose quartic term has a negative sign, which does not meet our goal.)

After this, let us examine under what value of $\lambda$ the only type of transformation that can meet our goal, the parallel shift of the ordinate axis given by (6) and (7), transforms the function (1) into the form (2). Substituting (6) and (7) into (1), we get

$$
v=\varphi(z)=(z+\lambda)^{4}+a(z+\lambda)^{3}+b(z+\lambda)^{2}+c(z+\lambda)+d
$$

Performing the powers and rearranging:

$$
v=z^{4}+(4 \lambda+a) z^{3}+\left(6 \lambda^{2}+3 \lambda a+b\right) z^{2}+\left(4 \lambda^{3}+3 \lambda^{2} a+2 \lambda b+c\right) z+\lambda^{4}+a \lambda^{3}+b \lambda^{2}+c \lambda+d
$$

This expression is in the form (2) if and only if the coefficients of the odd powers of $z$ are 0, i.e.,

$$
\begin{aligned}
4 \lambda+a & =0 \\
\text { hence } \lambda & =\frac{a}{4}
\end{aligned}
$$

and

$$
4 \lambda^{3}+3 \lambda^{2} a+2 \lambda b+c=0
$$

Substituting the value of $\lambda$ obtained from (9) into (10), we get:

$$
a^{3}-4 a b+8 c=0
$$

Thus, the necessary and sufficient condition for (1) to be expressible in the form (2) is that the coefficients of (1) satisfy the (11) relationship.

Since $\varphi(-z)=\varphi(z)$, the curve of the $\varphi$ function is symmetric with respect to the $v$ axis, so the fulfillment of the (11) condition, in other words, means that the image of the $f(x)$ function has a symmetry axis parallel to the $y$ axis.