(4x +( 5/ 15x2) + 7x-2) – (2x+(3 / 12x2) -7x -10) – (2x – (5 / 20x2) -29x +5) = 0

Let's simplify and solve the equation:

Given equation:

\[
(4x + \frac{5}{15x^2} + 7x - 2) - (2x + \frac{3}{12x^2} - 7x - 10) - (2x - \frac{5}{20x^2} - 29x + 5) = 0
\]

### Step 1: Simplify the terms inside each bracket.

For the first bracket:
\[
4x + \frac{5}{15x^2} + 7x - 2 = 11x + \frac{5}{15x^2} - 2
\]

For the second bracket:
\[
2x + \frac{3}{12x^2} - 7x - 10 = -5x + \frac{3}{12x^2} - 10
\]

For the third bracket:
\[
2x - \frac{5}{20x^2} - 29x + 5 = -27x - \frac{5}{20x^2} + 5
\]

### Step 2: Substitute the simplified expressions back into the equation.

\[
(11x + \frac{5}{15x^2} - 2) - (-5x + \frac{3}{12x^2} - 10) - (-27x - \frac{5}{20x^2} + 5) = 0
\]

### Step 3: Distribute the negative sign.

\[
11x + \frac{5}{15x^2} - 2 + 5x - \frac{3}{12x^2} + 10 + 27x + \frac{5}{20x^2} - 5 = 0
\]

### Step 4: Combine like terms.

For the \(x\) terms:
\[
11x + 5x + 27x = 43x
\]

For the constant terms:
\[
-2 + 10 - 5 = 3
\]

For the fractional terms:
\[
\frac{5}{15x^2} - \frac{3}{12x^2} + \frac{5}{20x^2}
\]

Let's simplify the fractions:
\[
\frac{5}{15x^2} = \frac{1}{3x^2}, \quad \frac{3}{12x^2} = \frac{1}{4x^2}, \quad \frac{5}{20x^2} = \frac{1}{4x^2}
\]

Now, combine them:
\[
\frac{1}{3x^2} - \frac{1}{4x^2} + \frac{1}{4x^2} = \frac{1}{3x^2}
\]

### Step 5: Form the simplified equation.

\[
43x + \frac{1}{3x^2} + 3 = 0
\]

### Step 6: Solve the equation.

To solve for \(x\), you need to clear the fraction by multiplying the entire equation by \(3x^2\):

\[
43x \cdot 3x^2 + 1 + 9x^2 = 0
\]

\[
129x^3 + 9x^2 + 1 = 0
\]

This is a cubic equation, which can be solved using numerical methods or factoring if possible. The exact solution would depend on further factorization or using a cubic equation solver.

Would you like to go through the solution process in more detail, or do you have any specific questions?

---

Here are five related questions:
1. How do you solve cubic equations using the factorization method?
2. What are the properties of cubic functions and their graphs?
3. How can you solve polynomial equations using the Rational Root Theorem?
4. How does the Discriminant help determine the nature of roots in cubic equations?
5. What is the significance of the coefficients in determining the shape of a cubic graph?

**Tip:** Always check for common factors before trying to solve complex equations—they can often simplify your work!

Learn how to solve the complex polynomial equation (4x + (5/15x^2) + 7x - 2) - (2x + (3/12x^2) - 7x - 10) - (2x - (5/20x^2) - 29x + 5) = 0 with detailed step-by-step instructions. This problem involves algebraic simplification and solving polynomial equations, suitable for advanced high school students and college-level learners.

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