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The expression in the image asks to simplify the following equation into a single fraction, with both the numerator and the denominator factored and simplified:

\[
\frac{12.5}{x} - 10 + \frac{4x}{2}, \quad \text{for } x \neq 0
\]

### Step 1: Simplify the terms
- The second term \( 10 \) is already simplified.
- The third term \( \frac{4x}{2} \) simplifies to \( 2x \).

Thus, the expression becomes:
\[
\frac{12.5}{x} - 10 + 2x
\]

### Step 2: Express all terms with a common denominator
To combine all terms into a single fraction, the common denominator is \( x \). The terms must be rewritten as:

- \( \frac{12.5}{x} \) remains the same.
- Rewrite \( 10 \) as \( \frac{10x}{x} \).
- Rewrite \( 2x \) as \( \frac{2x^2}{x} \).

Thus, the expression becomes:
\[
\frac{12.5}{x} - \frac{10x}{x} + \frac{2x^2}{x}
\]

### Step 3: Combine the terms
Now that the terms share the same denominator, combine the numerators:
\[
\frac{12.5 - 10x + 2x^2}{x}
\]

### Final Expression:
The simplified expression in a single fraction is:
\[
\frac{2x^2 - 10x + 12.5}{x}
\]

If you'd like further details or have any questions, feel free to ask!

### Related Questions:
1. How do you factor quadratic expressions like \( 2x^2 - 10x + 12.5 \)?
2. What steps should be taken if the denominator is a binomial instead of a single variable?
3. How would you simplify expressions with different denominators?
4. Can we apply this method to complex numbers or polynomials with higher degrees?
5. How does dividing by zero affect the expression?

### Tip:
When simplifying expressions, always look for a common denominator early to reduce the number of operations required later.

Écrire l'expression suivante sous la forme d'une seule fraction avec le numérateur et le dénominateur factorisés et simplifiés au maximum: 12.5/x - 10 + 4x/2 pour x ≠ 0

This problem involves simplifying and combining algebraic fractions into a single expression, with a focus on finding a common denominator and factoring. Suitable for students in Grades 10-12.

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