Given,
Equation of hyperbola,
16(x2+4x)−(y2−4y)+44=0
⇒16(x+2)2−64−(y−2)2+4+44=0
⇒16(x+2)2−(y−2)2=16
⇒1(x+2)2−16(y−2)2=1
Hence, equation of conjugate axis will be x=−2 and equation of transverse axis is given by y=2,
Now plotting the diagram of parabola x2=y+4 and x=-2&y=2 we get,

Now from above diagram, the area of the bounded region is given by,
A=∫−26(2−(x2−4))dx
⇒A=∫−26(6−x2)dx=(6x−3x3)−26
⇒A=(66−366)−(−12+38)
⇒A=3126+328
⇒A=46+328