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The image contains a mathematical equation involving partial fraction decomposition. Let's break down the steps visible from the image and work through solving the problem.

### Problem Breakdown:
The equation appears to involve the term:

\[
\frac{2s - 1}{s^2(s-1)}
\]

The goal seems to be to express this fraction using partial fraction decomposition. Here's the general idea for decomposition:

\[
\frac{2s - 1}{s^2(s - 1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s - 1}
\]

Where \( A \), \( B \), and \( C \) are constants we need to solve for.

### Step 1: Express the equation
We want to express the given fraction as:

\[
\frac{2s - 1}{s^2(s - 1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s - 1}
\]

Multiply both sides by the denominator \( s^2(s - 1) \) to get rid of the denominators:

\[
2s - 1 = A s(s - 1) + B(s - 1) + C s^2
\]

### Step 2: Expand the equation
Now, expand the right-hand side:

\[
A s(s - 1) = A(s^2 - s)
\]
\[
B(s - 1) = B(s - 1)
\]
\[
C s^2 = C s^2
\]

Combining these terms:

\[
2s - 1 = A(s^2 - s) + B(s - 1) + C s^2
\]
\[
2s - 1 = A s^2 - A s + B s - B + C s^2
\]
\[
2s - 1 = (A + C) s^2 + (-A + B) s - B
\]

### Step 3: Compare coefficients
Now, we compare the coefficients of \( s^2 \), \( s \), and the constant terms on both sides:

1. Coefficient of \( s^2 \): \( A + C = 0 \)
2. Coefficient of \( s \): \( -A + B = 2 \)
3. Constant term: \( -B = -1 \), so \( B = 1 \)

### Step 4: Solve the system of equations
From the third equation, \( B = 1 \).

Substitute \( B = 1 \) into the second equation:

\[
-A + 1 = 2 \implies A = -1
\]

Now substitute \( A = -1 \) into the first equation:

\[
A + C = 0 \implies -1 + C = 0 \implies C = 1
\]

### Final Result:
Thus, the partial fraction decomposition is:

\[
\frac{2s - 1}{s^2(s - 1)} = \frac{-1}{s} + \frac{1}{s^2} + \frac{1}{s - 1}
\]

Would you like further details on any of these steps or another explanation?

### 5 Related Questions:
1. How do you apply partial fraction decomposition to higher-degree polynomials?
2. What are the conditions required for partial fraction decomposition?
3. How would you decompose a fraction with repeated factors in the denominator?
4. Can you apply partial fraction decomposition in the context of Laplace transforms?
5. How do you solve systems of linear equations when comparing coefficients?

### Tip:
When working with partial fractions, always check if the degree of the numerator is lower than the degree of the denominator before proceeding with the decomposition. If not, perform polynomial division first.

Solve the equation involving partial fraction decomposition: \frac{2s - 1}{s^2(s - 1)}.

Step-by-step guide to solve the partial fraction decomposition of \frac{2s - 1}{s^2(s - 1)}. This detailed explanation includes expanding the equation and solving for constants A, B, and C. Ideal for high school students studying algebra and rational functions.

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