To find the value of the expression 2x^3 + 2x^2 - 7x + 72, we first need to determine the value of x.

Given that 2x = 3 + 5i, we can solve for x by dividing both sides of the equation by 2:

2x/2 = (3 + 5i)/2

Simplifying, we get:

x = (3 + 5i)/2

To further simplify, we can multiply the numerator and denominator by the conjugate of 2, which is 2 - 0i:

x = (3 + 5i)(2 - 0i)/(2)(2 - 0i)

Expanding the numerator, we get:

x = (6 + 10i - 0i - 0i^2)/(4 - 0i)

Simplifying, we have:

x = (6 + 10i)/(4)

Dividing each term by 4:

x = 6/4 + 10i/4

Simplifying further, we get:

x = 3/2 + 5i/2

Now that we have the value of x, we can substitute it into the expression 2x^3 + 2x^2 - 7x + 72.

Substituting x = 3/2 + 5i/2, we have:

2(3/2 + 5i/2)^3 + 2(3/2 + 5i/2)^2 - 7(3/2 + 5i/2) + 72

Now, let's simplify each term step by step.

1. Simplifying (3/2 + 5i/2)^3:
To simplify this, we can use the binomial expansion formula.

(3/2 + 5i/2)^3 = (3/2)^3 + 3(3/2)^2(5i/2) + 3(3/2)(5i/2)^2 + (5i/2)^3

Expanding each term, we get:

(27/8) + (27/4)(5i/2) + (9/2)(25i^2/4) + (125i^3/8)

Simplifying i^2 and i^3:

(27/8) + (135i/8) - (225i^2/8) - (125i/8)

Since i^2 = -1 and i^3 = -i, we have:

(27/8) + (135i/8) - (225(-1)/8) - (125(-i)/8)

Simplifying further, we get:

(27/8) + (135i/8) + (225/8) + (125i/8)

Combining like terms, we have:

(252/8) + (260i/8)

Simplifying, we get:

(63/2) + (65i/2)

2. Simplifying (3/2 + 5i/2)^2:
To simplify this, we can use the binomial expansion formula