solve for x: 2x(x^2 + 3)^3 - 8x^2 (x^2 + 3)^2 = 0

Let's solve the equation step by step.

Given equation:
\[
2x(x^2 + 3)^3 - 8x^2 (x^2 + 3)^2 = 0
\]

### Step 1: Factor out common terms
Both terms contain \( x(x^2 + 3)^2 \). Let's factor that out:

\[
x(x^2 + 3)^2 \left( 2(x^2 + 3) - 8x \right) = 0
\]

Now, we have two factors to solve:
1. \( x(x^2 + 3)^2 = 0 \)
2. \( 2(x^2 + 3) - 8x = 0 \)

### Step 2: Solve \( x(x^2 + 3)^2 = 0 \)
This gives two cases:
- \( x = 0 \)
- \( (x^2 + 3)^2 = 0 \), but \( x^2 + 3 = 0 \) has no real solutions because \( x^2 + 3 = 0 \) leads to \( x^2 = -3 \), which is impossible for real numbers.

So, from this factor, the solution is \( x = 0 \).

### Step 3: Solve \( 2(x^2 + 3) - 8x = 0 \)
Expand and solve:
\[
2x^2 + 6 - 8x = 0
\]
This simplifies to a quadratic equation:
\[
2x^2 - 8x + 6 = 0
\]
Now, divide through by 2 to simplify:
\[
x^2 - 4x + 3 = 0
\]
Factor the quadratic:
\[
(x - 3)(x - 1) = 0
\]
So, \( x = 3 \) or \( x = 1 \).

### Step 4: Final solutions
Combining all solutions, we have:
\[
x = 0, \, x = 1, \, x = 3
\]

### Final Answer:
\[
x = 0, 1, 3
\]

Would you like more details on any of the steps?

### Relative Questions:
1. How can we factor more complicated polynomial expressions?
2. What is the importance of factoring in solving equations?
3. How do you solve quadratic equations by factoring?
4. Why can't \( x^2 = -3 \) have real solutions?
5. What other methods exist for solving quadratic equations besides factoring?

### Tip:
Always check for common factors when solving polynomial equations—it simplifies the problem greatly!

Learn how to solve the polynomial equation 2x(x^2 + 3)^3 - 8x^2(x^2 + 3)^2 = 0 with a step-by-step solution. This problem involves key algebraic concepts, including polynomial equations, factoring, and solving quadratic equations. Suitable for students in Grades 9-11.

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