https://file.aiquickdraw.com/mathbot/file/172546064730585p7hxt0.jpg

The equation in the image appears to be:

\[ x^4 + 16x^2 + 256 \]

You are asked to factorize this expression.

### Factorization

Let's attempt to factorize the expression \( x^4 + 16x^2 + 256 \).

First, notice that this expression can be rewritten by substituting \( y = x^2 \):

\[
y^2 + 16y + 256
\]

Now, we need to factorize the quadratic expression \( y^2 + 16y + 256 \).

To factor this quadratic expression, we check if there are two numbers that multiply to \( 256 \) and add to \( 16 \). These numbers are \( 8 \) and \( 8 \):

\[
y^2 + 16y + 256 = (y + 8)(y + 8) = (y + 8)^2
\]

Substituting back \( y = x^2 \), we get:

\[
(x^2 + 8)^2
\]

Thus, the factorized form of \( x^4 + 16x^2 + 256 \) is:

\[
(x^2 + 8)^2
\]

Would you like more details on any part of this process?

### Follow-up Questions

1. What are the steps to factor a quadratic expression?
2. How do you recognize a perfect square trinomial?
3. What are the applications of factorizing polynomials?
4. How would you solve \( x^4 + 16x^2 + 256 = 0 \)?
5. How does substitution help in simplifying the factorization process?

**Tip:** Always check if a polynomial can be simplified or substituted to a more familiar form before factorizing it. This can often make the process much easier.

Learn how to factorize the polynomial x^4 + 16x^2 + 256 by recognizing it as a perfect square trinomial. This detailed step-by-step solution is suitable for students in Grades 9-12.

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