https://file.aiquickdraw.com/mathbot/file/17263165807729r1jti8q.jpg

Let’s break down the factorizations in the image. The problems seem to focus on factoring different algebraic expressions. I’ll go step by step for each one:

### 6.1 Factorize \( a^2 - 6ab + 9b^2 - 16c^2 \)
- The expression \( a^2 - 6ab + 9b^2 \) is a perfect square trinomial, which can be written as \( (a - 3b)^2 \).
- Now we have \( (a - 3b)^2 - 16c^2 \), which is a difference of squares: \( (A^2 - B^2) = (A - B)(A + B) \).
- So, the factorization becomes:
  \[
  (a - 3b + 4c)(a - 3b - 4c)
  \]
  **Answer:** \( (a - 3b + 4c)(a - 3b - 4c) \).

### 6.2 Factorize \( x^4 + x^2 + 1 \)
- We recognize this as a quadratic in disguise where \( u = x^2 \), so the expression becomes \( u^2 + u + 1 \).
- This doesn't factor nicely over the reals, but we can factor it over complex numbers using roots of unity.
- For simplicity, the factorization can be written as:
  \[
  (x^2 + x + 1)(x^2 - x + 1)
  \]
  **Answer:** \( (x^2 + x + 1)(x^2 - x + 1) \).

### 6.3 Factorize \( x^4 - 8x^2 + 4 \)
- Let \( u = x^2 \), so the expression becomes \( u^2 - 8u + 4 \), which is a quadratic.
- The quadratic formula gives:
 

Factorization of algebraic expressions including: a^2 - 6ab + 9b^2 - 16c^2, x^4 + x^2 + 1, x^4 - 8x^2 + 4, 8x^3 - 27y^3 - 4x + 6y, x^4 - 8x^2 - 33 - 14y^2, a^4/b^4 + 8a^2/b^2 + 36

Step-by-step guide on factorizing algebraic expressions such as a^2 - 6ab + 9b^2 - 16c^2 and x^4 + x^2 + 1. This problem covers algebraic factorization techniques, including the difference of squares and quadratics. Suitable for students in Grades 9-12.

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