This is a set of algebra factorization problems. I will solve a few examples step by step, then summarize and guide you further. Let’s start solving the problems one by one.

Solutions:

i. $2a^3 - 8a^2$

Step 1: Factor out the greatest common factor (GCF).
The GCF is $2a^2$.
$2a^3 - 8a^2 = 2a^2(a - 4)$
Final Answer: $2a^2(a - 4)$

ii. $\frac{1}{2}x^2 - \frac{3}{4}y$

Step 1: Factor out the GCF, which is $\frac{1}{4}$.
$\frac{1}{2}x^2 - \frac{3}{4}y = \frac{1}{4}(2x^2 - 3y)$
Final Answer: $\frac{1}{4}(2x^2 - 3y)$

iii. $2x^2 - 6xy$

Step 1: Factor out the GCF, which is $2x$.
$2x^2 - 6xy = 2x(x - 3y)$
Final Answer: $2x(x - 3y)$

iv. $2xy^3 + 6x^2y^2 + 8x^4y^4$

Step 1: Identify the GCF of all terms, which is $2xy^2$.
$2xy^3 + 6x^2y^2 + 8x^4y^4 = 2xy^2(y + 3x + 4x^3y^2)$
Final Answer: $2xy^2(y + 3x + 4x^3y^2)$

v. $x^2 - 4$

Step 1: Recognize this as a difference of squares.
$x^2 - 4 = (x - 2)(x + 2)$
Final Answer: $(x - 2)(x + 2)$

Would you like me to solve more of these problems or clarify any steps so far?

Follow-up Questions:

1. What is the general approach to factoring trinomials such as $x^2 + 7x + 12$?
2. How can we identify and factor a difference of squares like $x^2 - 4$?
3. Can you explain the process of factoring cubic expressions like $x^3 - 27$?
4. How do you handle factoring when the coefficients are fractions, such as in $\frac{1}{2}x^2 - \frac{3}{4}y$?
5. What are some tips for identifying the greatest common factor in complex expressions?

Tip:

Always start by factoring out the greatest common factor (GCF) before applying other methods such as grouping, difference of squares, or trinomials.