Answer
$\dfrac{25}{2}$
Work Step by Step
Let us consider that two continuous functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)$ on the interval $[a,b]$ . The area (A) of the region bounded by the graph of $f(x)$ and $g(x)$ on the interval $[a,b]$ can be calculated as: $Area(A)=\int_a^b [f(x)-g(x)] \ dx$
The given curves are $y=2(x+1)$ and $y=3(x=1)$.The intersection points of these curves satisfy the given equations are: $x=x^2-2 \implies 2(x+1)=3(x+1) \\ 2x+2=3x+30 \\ x=-1 \ \text{and}\ x=4$ . These points will be the lower and upper limit of the integration.
Thus, the area of the region is:
$A=\int_a^b [f(x)-g(x)] \ dx= \int_{-1}^4 [3(x+1)-2(x+1)] \ dx\\= \int_{-1}^4 (x+1)\ dx\\ =[\dfrac{x^2}{2}+x]_{-1}^4 \\=(\dfrac{16}{2}+4)-(\dfrac{1}{2}-1) \\=\dfrac{25}{2}$
