The given equation is:

\[
\frac{5x + 3}{3x - 2} - \frac{x - 1}{x - 3} = \frac{2x + 3}{x - 3}
\]

To solve this equation, let's go through it step by step.

### Step 1: Combine the fractions on the left-hand side
First, note that the fractions on the left-hand side have different denominators, so we need to find a common denominator. The common denominator for \((3x - 2)\) and \((x - 3)\) will be \((3x - 2)(x - 3)\).

Rewrite the left-hand side as:

\[
\frac{5x + 3}{3x - 2} - \frac{x - 1}{x - 3} = \frac{(5x + 3)(x - 3) - (x - 1)(3x - 2)}{(3x - 2)(x - 3)}
\]

### Step 2: Expand both numerators
Now, expand the numerators on both fractions.

For \((5x + 3)(x - 3)\):

\[
(5x + 3)(x - 3) = 5x^2 - 15x + 3x - 9 = 5x^2 - 12x - 9
\]

For \((x - 1)(3x - 2)\):

\[
(x - 1)(3x - 2) = 3x^2 - 2x - 3x + 2 = 3x^2 - 5x + 2
\]

### Step 3: Subtract the numerators
Now subtract the second expanded numerator from the first:

\[
(5x^2 - 12x - 9) - (3x^2 - 5x + 2) = 5x^2 - 12x - 9 - 3x^2 + 5x - 2 = 2x^2 - 7x - 11
\]

So the left-hand side becomes:

\[
\frac{2x^2 - 7x - 11}{(3x - 2)(x - 3)}
\]

### Step 4: Set up the equation
Now the equation becomes:

\[
\frac{2x^2 - 7x - 11}{(3x - 2)(x - 3)} = \frac{2x + 3}{x - 3}
\]

### Step 5: Cross multiply
Cross multiply both sides to eliminate the denominators:

\[
(2x^2 - 7x - 11)(x - 3) = (2x + 3)(3x - 2)
\]

### Step 6: Expand both sides
Expand both sides of the equation:

For the left-hand side:

\[
(2x^2 - 7x - 11)(x - 3) = 2x^3 - 6x^2 - 7x^2 + 21x - 11x + 33 = 2x^3 - 13x^2 + 10x + 33
\]

For the right-hand side:

\[
(2x + 3)(3x - 2) = 6x^2 - 4x + 9x - 6 = 6x^2 + 5x - 6
\]

### Step 7: Set the equation equal to zero
Now, set the two expressions equal to each other:

\[
2x^3 - 13x^2 + 10x + 33 = 6x^2 + 5x - 6
\]

Move all terms to one side of the equation:

\[
2x^3 - 13x^2 + 10x + 33 - 6x^2 - 5x + 6 = 0
\]

Simplify:

\[
2x^3 - 19x^2 + 5x + 39 = 0
\]

### Step 8: Solve the cubic equation
We now need to solve the cubic equation:

\[
2x^3 - 19x^2 + 5x + 39 = 0
\]

This can be solved using methods such as synthetic division or numerical methods. 

Would you like me to solve the cubic equation in more detail or give a numerical approximation? 

---

### Questions to expand your understanding:
1. How do we solve cubic equations in general?
2. What are common methods to find rational roots of polynomials?
3. Why do we use common denominators when subtracting fractions?
4. How can we simplify a complex fraction like the one in this problem?
5. What are the steps to cross-multiply fractions in an equation?

#### Tip:
Whenever working with fractions, it’s always helpful to factorize and simplify as much as possible to avoid unnecessary complications.

5x + 3 / 3x - 2 - x - 1 / x - 3 = 2x + 3 / x - 3

Learn how to solve the rational equation (5x + 3)/(3x - 2) - (x - 1)/(x - 3) = (2x + 3)/(x - 3) using algebraic techniques. This problem involves finding common denominators, expanding expressions, and solving a cubic equation. Suitable for high school students in Grades 10-12.

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