```
(25 points) Let $x, y, z, r$ be real numbers. Find the minimum value of the following algebraic expressions:
$$
\begin{array}{l}
\text { (1) } x^{2}+y^{2}+x y+x+y \text {; } \\
\text { (2) } x^{2}+y^{2}+z^{2}+x y+y z+z x+x+y+z \text {; } \\
\text { (3) } x^{2}+y^{2}+z^{2}+r^{2}+x y+x z+x r+y z+y r+ \\
z r+x+y+z+r
\end{array}
$$
```
$$
\begin{array}{l}
\text { (1) } x^{2}+y^{2}+x y+x+y \\
=\left(x+\frac{y+1}{2}\right)^{2}+\frac{3}{4}\left(y+\frac{1}{3}\right)^{2}-\frac{1}{3} .
\end{array}
$$

When $x=y=-\frac{1}{3}$, the minimum value of the given algebraic expression is $-\frac{1}{3}$.
$$
\begin{array}{l}
\text { (2) } x^{2}+y^{2}+z^{2}+x y+z x+y z+x+y+z \\
=\left(x+\frac{y+z+1}{2}\right)^{2}+\frac{3}{4}\left(y+\frac{z+1}{3}\right)^{2}+ \\
\frac{2}{3}\left(z+\frac{1}{4}\right)^{2}-\frac{3}{8} \text {. } \\
\end{array}
$$

When $x=y=z=-\frac{1}{4}$, the minimum value of the given algebraic expression is $-\frac{3}{8}$.
(3) When $x=y=z=r=-\frac{1}{5}$, the minimum value of the given algebraic expression is
$$
4\left(-\frac{1}{5}\right)^{2}+6\left(-\frac{1}{5}\right)\left(-\frac{1}{5}\right)+4\left(-\frac{1}{5}\right)=-\frac{2}{5} \text {. }
$$