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To solve the given equation by factorization, let's analyze and simplify it step by step.

The equation provided is:

\[
\frac{5}{x - 2} = 2 - \frac{4}{(x - 2)^2}
\]

### Step 1: Eliminate the fractions by multiplying through by \((x - 2)^2\).

Multiply every term by \((x - 2)^2\) to get rid of the denominators:

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

### Step 2: Expand both sides.

Expand the expressions on both sides:

- Left side: \(5(x - 2) = 5x - 10\)
- Right side: \(2(x - 2)^2 = 2(x^2 - 4x + 4)\)

So,

\[
5x - 10 = 2(x^2 - 4x + 4) - 4
\]

### Step 3: Simplify the right side.

Expand and simplify:

\[
5x - 10 = 2x^2 - 8x + 8 - 4
\]

\[
5x - 10 = 2x^2 - 8x + 4
\]

### Step 4: Rearrange the equation.

Bring all terms to one side of the equation:

\[
0 = 2x^2 - 8x - 5x + 4 + 10
\]

Combine like terms:

\[
0 = 2x^2 - 13x + 14
\]

### Step 5: Factorize the quadratic equation.

Now, factorize \(2x^2 - 13x + 14\). We need to find two numbers that multiply to \(2 \times 14 = 28\) and add to \(-13\). The numbers are \(-7\) and \(-6\).

So, we factorize as:

\[
2x^2 - 7x - 6x + 14 = 0
\]

Group the terms:

\[
(2x^2 - 7x) - (6x - 14) = 0
\]

Factor out the common factors:

\[
x(2x - 7) - 2(3x - 7) = 0
\]

The expression can be factorized as:

\[
(2x - 7)(x - 2) = 0
\]

### Step 6: Solve for \(x\).

Set each factor to zero and solve for \(x\):

\[
2x - 7 = 0 \quad \text{or} \quad x - 2 = 0
\]

\[
x = \frac{7}{2} \quad \text{or} \quad x = 2
\]

However, since \(x = 2\) would make the original equation undefined (denominator would be zero), we discard \(x = 2\).

### Final Answer:

The solution is:

\[
x = \frac{7}{2}
\]

Would you like more details on any step, or do you have any questions?

---

Here are five relative questions to expand your understanding:

1. How do you determine which values of \(x\) to discard when solving rational equations?
2. What is the significance of factoring in solving quadratic equations?
3. How does multiplying through by the common denominator help simplify equations with fractions?
4. Can you solve similar equations where the degree of the polynomial in the denominator is higher?
5. What techniques can you use if the quadratic equation is not easily factorable?

**Tip:** Always check the solution in the original equation to ensure it's valid, especially when dealing with rational expressions!

Learn how to factorize the rational equation 5/(x - 2) = 2 - 4/(x - 2)^2 with a step-by-step solution. This problem covers key concepts in equations, rational equations, and quadratic equations. Suitable for advanced high school students.

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