Let's calculate the minimum value of the following expression, as well as its maximum value between -2 and -1.

$$
x(x+1)(x+2)(x+3)
$$
I. First, by forming the product of the two extreme and the two inner factors, and denoting the first product by $z$

$$
K=\left(x^{2}+3 x\right)\left(x^{2}+3 x+2\right)=z(z+2)=(z+1)^{2}-1
$$

Accordingly, the smallest value of $K$ is obtained when $z+1=0$, and in this case $K_{\min }=-1$. (A value of $x$ that leads to this is because

$$
\begin{gathered}
z+1=x^{2}+3 x+1=0 \text {, from which } \\
\left.x^{\prime}=\frac{-3-\sqrt{5}}{2}=-2.618, \quad x^{\prime \prime}=-0.382 .\right)
\end{gathered}
$$

II. The value of $K$ at $x=0, -1, -2, -3$ is 0, because at each of these points one of its factors is 0, so the curve representing $K$ intersects the $x$-axis at these four points.

![](https://cdn.mathpix.com/cropped/2024_05_02_73c0104a3418fafb2d8fg-1.jpg?height=582&width=753&top_left_y=663&top_left_x=686)

The four points are pairwise symmetric with respect to the point $C$ with abscissa $-3/2$, from which we conjecture that the curve is symmetric with respect to the line parallel to the $y$-axis passing through $C$. Indeed, if we measure the abscissas of the points from $C$ instead of the origin, and denote this by $u$

$$
\begin{gathered}
u=x+\frac{3}{2}, \quad \text { i.e., } \quad x=u-\frac{3}{2} \text {, and thus } \\
K=\left(u-\frac{3}{2}\right)\left(u-\frac{1}{2}\right)\left(u+\frac{1}{2}\right)\left(u+\frac{3}{2}\right)=\left(u^{2}-\frac{9}{4}\right)\left(u^{2}-\frac{1}{4}\right)=u^{2}\left(u^{2}-\frac{5}{2}\right)+\frac{9}{16}
\end{gathered}
$$

from which we see that the value of $K$ at $u$ and $-u$ is the same, confirming the conjectured symmetry.

Now, in the domain of interest

$$
\begin{gathered}
-2 \leq x=u-\frac{3}{2} \leq-1, \text { so } \\
-\frac{1}{2} \leq u \leq \frac{1}{2}, \quad \text { and thus } \quad u^{2} \leq \frac{1}{4}
\end{gathered}
$$

and of course $u^{2} \geq 0$. Accordingly, the second factor on the right-hand side of (1) is negative at all points of interest, while the first factor is positive—except at $u=0$, where it is 0. Thus, the product is negative or 0, so its maximum value is $K_{\max }=9 / 16$, at $u=0$, or $x=-3 / 2$.