To solve the equation \(2x^2 + 3x - 2 = (2x - 1)\sqrt{2x^2 + x - 3}\), we need to isolate the square root term and then square both sides to eliminate the square root. Here are the steps:

1. **Isolate the square root term:**
\[
2x^2 + 3x - 2 = (2x - 1)\sqrt{2x^2 + x - 3}
\]

2. **Square both sides to eliminate the square root:**
\[
(2x^2 + 3x - 2)^2 = \left((2x - 1)\sqrt{2x^2 + x - 3}\right)^2
\]

Simplifying the right side:
\[
(2x^2 + 3x - 2)^2 = (2x - 1)^2 (2x^2 + x - 3)
\]

3. **Expand both sides:**

Left side:
\[
(2x^2 + 3x - 2)^2 = (2x^2 + 3x - 2)(2x^2 + 3x - 2)
\]
\[
= 4x^4 + 6x^3 - 4x^2 + 6x^3 + 9x^2 - 6x - 4x^2 - 6x + 4
\]
\[
= 4x^4 + 12x^3 + x^2 - 12x + 4
\]

Right side:
\[
(2x - 1)^2 (2x^2 + x - 3) = (4x^2 - 4x + 1)(2x^2 + x - 3)
\]
\[
= 4x^2(2x^2 + x - 3) - 4x(2x^2 + x - 3) + 1(2x^2 + x - 3)
\]
\[
= 8x^4 + 4x^3 - 12x^2 - 8x^3 - 4x^2 + 12x + 2x^2 + x - 3
\]
\[
= 8x^4 - 12x^2 + 12x + x - 3
\]
\[
= 8x^4 - 12x^2 + 13x - 3
\]

4. **Set the expanded forms equal to each other:**
\[
4x^4 + 12x^3 + x^2 - 12x + 4 = 8x^4 - 12x^2 + 13x - 3
\]

5. **Move all terms to one side to set the equation to zero:**
\[
4x^4 + 12x^3 + x^2 - 12x + 4 - 8x^4 + 12x^2 - 13x + 3 = 0
\]
\[
-4x^4 + 12x^3 + 13x^2 - 25x + 7 = 0
\]

6. **Solve the polynomial equation:**
Solving a fourth-degree polynomial analytically can be quite complex and often requires numerical methods or factoring techniques if possible. However, we can check for rational roots using the Rational Root Theorem or use numerical methods such as Newton's method to approximate the roots.

Given the complexity, let's check for possible rational roots using the Rational Root Theorem, which states that any rational root, in the form of \(\frac{p}{q}\), is a factor of the constant term (7) divided by a factor of the leading coefficient (-4).

Possible rational roots are: \(\pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{1}{4}, \pm \frac{7}{4}\).

Testing these values in the polynomial can help identify any rational roots. If no rational roots are found, numerical methods or graphing tools can be used to approximate the solutions.