Step 1:
Given $y=-x^2$ and the line $x+y+2=0$ (i.e) $y=-(x+2)$
Consider the curve $y=-x^2$.Clearly this curve is a parabola open downwards,with vertex (0,0).
To obtain the points of intersection,let us solve the equation,by equating them.
$-x^2=-(x+2)$
$\Rightarrow x^2-x-2=0$
On factorising we get,
$(x-2)(x-1)=0.$
(i.e) $x=2$ or $-1$ and $y=-4$ or $-1$
Hence the points of intersection are $(2,-4)$ and $(-1,-1)$.
Step 2:
The required area is the shaded region shown in the fig below.
The area of the shaded portion $A=\int_{-1}^2(y_2-y_1)dx.$
Where $y_2=-x^2$ and $y_1=-(x+2)$
$A=\int_{-1}^2(-x^2)dx-\int_{-1}^2-(x+2)dx.$
On integrating we get,
$\;\;=-\begin{bmatrix}\large\frac{x^3}{3}\end{bmatrix}_{-1}^2$$+\begin{bmatrix}\big(\large\frac{x^2}{2}\big)\normalsize+2x\end{bmatrix}_{-1}^2$
Step 3:
On applying limits we get,
$\;\;=-\begin{bmatrix}\large\frac{2^3}{3}-\large\frac{(-1)^3}{3}\end{bmatrix}+\begin{bmatrix}\large\frac{2^2}{2}-\frac{(-1)^2}{2}\normalsize+2(2)-2(-1)\end{bmatrix}$
$\;\;=-\begin{bmatrix}\large\frac{8}{3}+\frac{1}{3}\end{bmatrix}+\begin{bmatrix}\large\frac{4}{2}-\frac{1}{2}+\normalsize 4+2\end{bmatrix}$
$\;\;=[-3]+[8-\large\frac{1}{2}]$
$\;\;=\large\frac{17}{2}$$-3=\large\frac{9}{2}$
Hence the required area is $\large\frac{9}{2}$sq.units.