Question: Simplify $x \times 3 - \frac{1000}{x} \times 2 - 8x + 20$
1. **State the problem:** Simplify the expression $x \times 3 - \frac{1000}{x} \times 2 - 8x + 20$.
2. **Rewrite the expression clearly:**
$$3x - \frac{2000}{x} - 8x + 20$$
3. **Combine like terms:**
$$3x - 8x = -5x$$
So the expression becomes:
$$-5x - \frac{2000}{x} + 20$$
4. **Find a common denominator to combine terms:**
The terms $-5x$ and $20$ can be written with denominator $x$:
$$-5x = -\frac{5x^2}{x}$$
$$20 = \frac{20x}{x}$$
5. **Rewrite entire expression with denominator $x$:**
$$-\frac{5x^2}{x} - \frac{2000}{x} + \frac{20x}{x} = \frac{-5x^2 - 2000 + 20x}{x}$$
6. **Rearrange numerator:**
$$\frac{-5x^2 + 20x - 2000}{x}$$
7. **Factor numerator:**
Factor out $-5$:
$$\frac{-5(x^2 - 4x + 400)}{x}$$
8. **Check if quadratic $x^2 - 4x + 400$ factors further:**
Calculate discriminant:
$$\Delta = (-4)^2 - 4 \times 1 \times 400 = 16 - 1600 = -1584 < 0$$
No real factors.
9. **Final simplified form:**
$$\boxed{\frac{-5(x^2 - 4x + 400)}{x}}$$
This is the simplest form with real coefficients.