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Let's solve the given equation step by step.

The equation is:
\[
6x^3(2x + 4) = 2x(6x^3 + 12x^2 - 10) + 60
\]

### Step 1: Expand both sides

On the left-hand side:
\[
6x^3(2x + 4) = 6x^3 \cdot 2x + 6x^3 \cdot 4 = 12x^4 + 24x^3
\]

On the right-hand side:
\[
2x(6x^3 + 12x^2 - 10) + 60 = 2x \cdot 6x^3 + 2x \cdot 12x^2 + 2x \cdot (-10) + 60
\]
\[
= 12x^4 + 24x^3 - 20x + 60
\]

### Step 2: Set both sides equal

Now we have:
\[
12x^4 + 24x^3 = 12x^4 + 24x^3 - 20x + 60
\]

### Step 3: Simplify the equation

Subtract \(12x^4 + 24x^3\) from both sides:
\[
0 = -20x + 60
\]

### Step 4: Solve for \(x\)

Now we solve:
\[
20x = 60
\]
\[
x = 3
\]

### Conclusion

The solution is \(x = 3\).

The correct answer from the options provided is \(x = 3\).

Do you want more details, or do you have any questions? Here are some related questions to further your understanding:

1. How do you expand polynomial expressions involving multiplication?
2. Why does subtracting the same terms from both sides simplify an equation?
3. How would the solution change if there was an additional \(x^2\) term on the left side?
4. How do you solve an equation with multiple \(x^n\) terms, like \(x^4\) or \(x^3\)?
5. What methods can be used to check the solution of a polynomial equation?

**Tip:** Always double-check each step in simplifying and expanding, especially when dealing with higher-degree polynomials.

What is the solution to the equation 6x^3(2x + 4) = 2x(6x^3 + 12x^2 - 10) + 60?

Learn how to solve the polynomial equation 6x^3(2x + 4) = 2x(6x^3 + 12x^2 - 10) + 60 with a detailed step-by-step solution. This problem covers key algebraic concepts, including polynomial expansion, simplification, and solving linear equations. Suitable for students in Grades 10-12.

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