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We are given the equation:

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

Let's solve this step-by-step.

### Step 1: Cross multiply
To eliminate the fractions, we'll cross multiply:

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

### Step 2: Expand both sides
Now, expand both sides of the equation:

Left side:
\[
(3w + 2)(w - 1) = 3w(w - 1) + 2(w - 1) = 3w^2 - 3w + 2w - 2 = 3w^2 - w - 2
\]

Right side:
\[
w(w - 4) = w^2 - 4w
\]

So, the equation becomes:
\[
3w^2 - w - 2 = w^2 - 4w
\]

### Step 3: Move all terms to one side
Let's move everything to the left side:

\[
3w^2 - w - 2 - w^2 + 4w = 0
\]

Simplify:
\[
2w^2 + 3w - 2 = 0
\]

### Step 4: Solve the quadratic equation
We can now solve this quadratic equation using the quadratic formula:

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

For the equation \( 2w^2 + 3w - 2 = 0 \), we have:
- \(a = 2\)
- \(b = 3\)
- \(c = -2\)

Substitute these into the quadratic formula:

\[
w = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}
\]
\[
w = \frac{-3 \pm \sqrt{9 + 16}}{4}
\]
\[
w = \frac{-3 \pm \sqrt{25}}{4}
\]
\[
w = \frac{-3 \pm 5}{4}
\]

### Step 5: Find the two possible solutions
Now, compute the two possible values for \(w\):

1. \(w = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}\)
2. \(w = \frac{-3 - 5}{4} = \frac{-8}{4} = -2\)

### Step 6: Check for restrictions
We must check if these solutions are valid by ensuring they don't make the denominators zero.

For \(w = \frac{1}{2}\), the denominator \(w - 1 = \frac{1}{2} - 1 = -\frac{1}{2}\), which is non-zero.

For \(w = -2\), the denominator \(w - 1 = -2 - 1 = -3\), which is also non-zero.

Both solutions are valid.

### Final Answer:
The solutions are \(w = \frac{1}{2}\) and \(w = -2\).

---

Would you like more details on any part of the solution or have further questions?

Here are 5 related questions to expand on this problem:
1. How do you factor quadratic equations instead of using the quadratic formula?
2. How do you handle equations with multiple fractions?
3. What are the conditions for using the quadratic formula?
4. How would the solution change if there were no real roots?
5. What are the different ways to check if a solution is valid?

**Tip:** Always check the denominators when solving equations with fractions to avoid division by zero errors.

Solve the equation: (3w + 2)/w = (w - 4)/(w - 1)

Learn how to solve a rational equation involving cross multiplication and a quadratic formula. This detailed step-by-step guide is designed for students in Grades 9-11 and covers algebraic concepts including rational expressions and solving quadratic equations.

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