We are given the equation:

\[
\frac{2x + 3}{x - 1} = \frac{4x - 1}{x + 2}
\]

To solve this, follow these steps:

### Step 1: Cross-multiply to eliminate the fractions.

Cross-multiplying gives:

\[
(2x + 3)(x + 2) = (4x - 1)(x - 1)
\]

### Step 2: Expand both sides of the equation.

Expanding the left side:

\[
(2x + 3)(x + 2) = 2x(x + 2) + 3(x + 2) = 2x^2 + 4x + 3x + 6 = 2x^2 + 7x + 6
\]

Expanding the right side:

\[
(4x - 1)(x - 1) = 4x(x - 1) - 1(x - 1) = 4x^2 - 4x - x + 1 = 4x^2 - 5x + 1
\]

### Step 3: Set up the equation by combining both expanded sides.

Now we have:

\[
2x^2 + 7x + 6 = 4x^2 - 5x + 1
\]

### Step 4: Bring all terms to one side of the equation.

Move all terms to the left side:

\[
2x^2 + 7x + 6 - 4x^2 + 5x - 1 = 0
\]

Simplifying the equation:

\[
-2x^2 + 12x + 5 = 0
\]

### Step 5: Solve the quadratic equation.

We now have the quadratic equation:

\[
-2x^2 + 12x + 5 = 0
\]

Dividing through by -1 to make the leading coefficient positive:

\[
2x^2 - 12x - 5 = 0
\]

We can solve this using the quadratic formula. The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where \(a = 2\), \(b = -12\), and \(c = -5\).

Substituting these values into the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(2)(-5)}}{2(2)}
\]
\[
x = \frac{12 \pm \sqrt{144 + 40}}{4}
\]
\[
x = \frac{12 \pm \sqrt{184}}{4}
\]
\[
x = \frac{12 \pm \sqrt{4 \times 46}}{4}
\]
\[
x = \frac{12 \pm 2\sqrt{46}}{4}
\]
\[
x = \frac{12}{4} \pm \frac{2\sqrt{46}}{4}
\]
\[
x = 3 \pm \frac{\sqrt{46}}{2}
\]

### Step 6: Final Solutions

The two possible solutions are:

\[
x = 3 + \frac{\sqrt{46}}{2} \quad \text{or} \quad x = 3 - \frac{\sqrt{46}}{2}
\]

These are the exact solutions to the given equation.

---

Would you like more details on any of the steps, or have any other questions? 

Here are 5 related questions to think about:
1. How do you cross-multiply when solving rational equations?
2. What are some strategies for factoring quadratic equations?
3. Can quadratic equations always be solved by factoring?
4. What does the discriminant tell you about the nature of the solutions?
5. How do you simplify radicals like \( \sqrt{184} \)?

**Tip:** When using the quadratic formula, always check the discriminant \(b^2 - 4ac\) first to determine the number and type of solutions (real or complex).

This page provides a detailed step-by-step solution to the rational equation (2x + 3) / (x - 1) = (4x - 1) / (x + 2), covering key algebraic concepts such as cross-multiplication, expanding equations, and solving quadratics using the quadratic formula. Suitable for Grades 9-11 students.

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