To find the area bounded by the parabola y^2 = 4x and the line x + y = 3, we need to find the points of intersection between the parabola and the line and then calculate the area between them.

Finding the Points of Intersection:
To find the points of intersection, we can set the equations of the parabola and the line equal to each other and solve for x and y:

y^2 = 4x
x + y = 3

Substituting y = 3 - x into the equation of the parabola, we get:

(3 - x)^2 = 4x

Expanding and rearranging the equation, we get:

9 - 6x + x^2 = 4x
x^2 + 10x - 9 = 0

Now, we can solve this quadratic equation for x using factoring or the quadratic formula. By factoring, we can write it as:

(x + 9)(x - 1) = 0

This gives us two possible values for x: x = -9 and x = 1.

Substituting these values back into the equation of the line, we can find the corresponding y-values:

For x = -9:
y = 3 - (-9) = 3 + 9 = 12

For x = 1:
y = 3 - 1 = 2

So, the points of intersection are (-9, 12) and (1, 2).

Calculating the Area:
To calculate the area bounded by the parabola and the line, we can integrate the difference between the y-values of the parabola and the line over the interval between the x-values of intersection points.

We need to find the integral of (y - (3 - x)) with respect to x, where y is given by the equation of the parabola: y = √(4x).

The integral becomes:

∫[(√(4x) - (3 - x))] dx

Integrating and evaluating this integral from x = -9 to x = 1 gives us the required area:

A = ∫[(√(4x) - (3 - x))] dx (from x = -9 to x = 1)

Simplifying the integral and evaluating it, we get:

A = [(2/3)x^(3/2) + (x^2)/2 - 3x] (from x = -9 to x = 1)
= [2/3 + 1/2 - 3 - ((2/3)(-9)^(3/2) + ((-9)^2)/2 - 3(-9))]

Calculating this expression, we get:

A = [2/3 + 1/2 - 3 - (2/3 - 243/2 + 27)]
= [2/3 + 1/2 - 3 - 2/3 + 243/2 - 27]
= [16/3]

Therefore, the area bounded by the parabola y^2 = 4x and the line x + y = 3 is 16/3.