Find the area of the region bounded by the graphs of the algebraic functions: 𝑓(𝑥)=𝑥2+6𝑥+9 and  𝑔(𝑥)=5(𝑥+3).

To find the area between two curves, we need to integrate the absolute difference of the two functions over the interval where they intersect.

Step 1: Set the two functions equal to each other to find the points of intersection.

x^2 + 6x + 9 = 5(x + 3)

Solving this equation gives us x = -3. So, the two functions intersect at x = -3.

Step 2: Find the area between the curves.

The area A between the two curves from x = a to x = b is given by the formula:

A = ∫ from a to b |f(x) - g(x)| dx

In this case, f(x) = x^2 + 6x + 9 and g(x) = 5(x + 3), and the interval is from -3 to -3.

However, since the interval is just a single point, the area between the curves is 0.

So, the area of the region bounded by the graphs of the algebraic functions f(x) = x^2 + 6x + 9 and g(x) = 5(x + 3) is 0.