88 In the Cartesian coordinate system $x O y$, the area of the figure formed by points whose coordinates satisfy the condition $\left(x^{2}+y^{2}+2 x+2 y\right)\left(4-x^{2}-y^{2}\right) \geqslant$ 0 is
$882 \pi+4$.
$$
\left\{\begin{array} { l } 
{ x ^ { 2 } + y ^ { 2 } + 2 x + 2 y \geqslant 0 } \\
{ 4 - ( x ^ { 2 } + y ^ { 2 } ) \geqslant 0 }
\end{array} \text { or } \left\{\begin{array}{l}
x^{2}+y^{2}+2 x+2 y \leqslant 0 \\
4-\left(x^{2}+y^{2}\right) \leqslant 0
\end{array}\right.\right.
$$

The shape formed is the shaded part in Figure 10. The area of the segment of the large circle with the base at $(-2,0),(0,-2)$ is
$$
S_{\text {segment }}=\frac{1}{2} \times 2^{2} \times \frac{\pi}{2}-\frac{1}{2} \times 2^{2}=\pi-2 .
$$

The required area is
$$
S_{\text {large }}-2 S_{\text {segment }}=4 \pi-2(\pi-2)=2 \pi+4
$$