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: