To divide the polynomial \( 4x^4 - 12x^3 + 10x^2 - 6x + 4 \) by \( x - 2 \), we can use **synthetic division** or **long division**. I'll apply synthetic division in this case.

### Synthetic Division Steps:
1. Write the coefficients of the polynomial:  
   \( [4, -12, 10, -6, 4] \)
2. Use the root of \( x - 2 = 0 \), so \( x = 2 \).
3. Set up the synthetic division process:
   - Multiply, add, repeat.

The division process:

\[
\begin{array}{r|rrrrr}
2 & 4 & -12 & 10 & -6 & 4 \\
  &   & 8  & -8 & 4  & -4 \\
\hline
  & 4 & -4 & 2  & -2 & 0 \\
\end{array}
\]

So, the quotient is:

\[
4x^3 - 4x^2 + 2x - 2
\]

And since the remainder is 0, the final result is:
\[
\boxed{4x^3 - 4x^2 + 2x - 2}
\]

Would you like more details on any of these steps, or help with a related question?

### Related Questions:
1. How does synthetic division differ from long division?
2. How do you handle remainders in polynomial division?
3. Can synthetic division be used for divisors other than \(x - a\)?
4. What are the practical applications of polynomial division?
5. How do you factor a polynomial after dividing it?

**Tip:** Synthetic division only works for divisors in the form \(x - a\), where \(a\) is a constant.

Learn how to divide the polynomial 4x^4 - 12x^3 + 10x^2 - 6x + 4 by x - 2 using synthetic division. This detailed step-by-step solution covers polynomial division and key algebraic concepts. Suitable for students in Grades 10-12.

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