Let's solve the equation step by step:

Given equation:

\[
4x + 3x^3 - \left( 17x^2 + 2 \right)^2 x^5 = 0
\]

### Step 1: Expand \( \left( 17x^2 + 2 \right)^2 \)

We need to first expand the square term:

\[
\left( 17x^2 + 2 \right)^2 = \left( 17x^2 \right)^2 + 2(17x^2)(2) + 2^2
\]

\[
= 289x^4 + 68x^2 + 4
\]

### Step 2: Substitute the expanded term into the equation

Now replace \( \left( 17x^2 + 2 \right)^2 \) with \( 289x^4 + 68x^2 + 4 \) in the original equation:

\[
4x + 3x^3 - \left( 289x^4 + 68x^2 + 4 \right) x^5 = 0
\]

### Step 3: Distribute the \( x^5 \) term

Distribute \( x^5 \) across each term inside the parentheses:

\[
4x + 3x^3 - (289x^9 + 68x^7 + 4x^5) = 0
\]

### Step 4: Rearrange the equation

Now, we have:

\[
4x + 3x^3 - 289x^9 - 68x^7 - 4x^5 = 0
\]

### Step 5: Combine like terms (if possible)

In this case, no terms can be combined. Therefore, the final equation is:

\[
4x + 3x^3 - 289x^9 - 68x^7 - 4x^5 = 0
\]

This is the simplified form of the given equation. You would now proceed to solve this either by factoring or numerical methods (depending on the degree and complexity). Factoring doesn't seem straightforward here, so numerical methods like Newton's method might be a practical approach.

Would you like to see more details or help with a specific solution method?

### Related Questions:
1. How do you apply Newton's method to solve higher-degree polynomials?
2. What is the significance of the degree of a polynomial in solving equations?
3. How can we factor high-degree polynomials more effectively?
4. Can graphing the equation help find approximate solutions?
5. How does multiplying polynomials impact the degree of the equation?

### Tip:
When solving complex equations, it's useful to first expand and simplify as much as possible to spot patterns or potential factorable terms.

This detailed guide walks through solving the polynomial equation 4x + 3x³ - (17x² + 2)²x⁵ = 0. Learn the step-by-step algebraic expansion and simplification techniques. Suitable for students in Grades 10-12 studying advanced algebra.

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