$\left[\begin{array}{l}\text { Substitution of variables (other) }\end{array}\right]$
$\left[\begin{array}{l}\text { Completing the square. Sums of squares }\end{array}\right]$
$\left[\begin{array}{l}\text { Polynomials (other) }\end{array}\right]$

Given the polynomial $x(x+1)(x+2)(x+3)$. Find its minimum value.
$x(x+3)(x+1)(x+2)=\left(x^{2}+3 x\right)\left(x^{2}+3 x+2\right)$. Let's denote $x^{2}+3 x$ by $z$. Then $\left(x^{2}+3 x\right)\left(x^{2}+3 x+2\right)=z(z+2)=(z+1)^{2}-1$. The minimum value -1 of this function is reached when $z=-1$. The equation $x^{2}+3 x+1=0$ has solutions (the discriminant is greater than zero), therefore, such an $x$ exists for which our function reaches the value 1.

## Answer

$-1$.