Final answer:
The quadratic function that models the area of the framed painting is A(x) = 4x^2 + 80x + 319, derived by considering the painting's dimensions with added frame width.
Step-by-step explanation:
To derive the quadratic function that models the area of the painting including its frame, we need to add the width of the frame to all sides of the painting dimensions. The painting is 29 inches by 11 inches, and each side of the frame adds 'x' inches to these dimensions, making the total dimensions of the framed painting (29+2x) inches by (11+2x) inches.
The area of the painting and frame is found by multiplying the length by the width:
Area = (29 + 2x)(11 + 2x) = 319 + 58x + 22x + 4x^2 = 4x^2 + 80x + 319.
Therefore, the quadratic function that represents the area of the painting and frame is A(x) = 4x^2 + 80x + 319.