Let's break down the expressions from the image step by step:

1. Expression 1:

$5ax - 5a^2x^2$ This expression contains two terms involving $a$, $x$, and $x^2$. It can be factored as: $5a(x - ax^2)$ This factors out the common factor $5a$.

2. Expression 2:

$3a^4 - 3a^2b + 6a^2b$ Here we can combine like terms. The second and third terms involve $a^2b$, so: $3a^4 + (-3a^2b + 6a^2b) = 3a^4 + 3a^2b$ This expression is simplified as: $3a^2(a^2 + b)$

3. Expression 3:

$mx + 2my - nx - 2ny$ Here we group the terms involving $x$ and $y$: $(mx - nx) + (2my - 2ny)$ Factor out the common terms: $(m-n)x + 2(m-n)y$

4. Expression 4:

$x^4 + x^3 + 2x + 2$ This is a polynomial with four terms. It doesn't appear to have any obvious factorization, but it can be grouped or used as part of a broader algebraic or calculus problem (such as factoring or finding roots). This can be left as is unless further instructions require manipulation.

Would you like further details or to ask any questions about these steps?

Here are 5 related questions:

1. How would you solve for $x$ in the second expression if it's part of an equation?
2. Can the fourth expression be factored any further?
3. How would you apply factoring techniques for polynomials in a real-world application?
4. Can you express the first equation in terms of another variable?
5. What happens if you substitute specific values for $a$ and $b$ in the second expression?

Tip: Always look for common factors across terms in an expression; this helps simplify complex equations effectively.