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Let's go through the problems step by step and factor them.

### 1. \( 50p^2q + 35p^3q^2 \)

We can start by identifying the common factors between both terms:
- The GCF of the coefficients 50 and 35 is 5.
- The common factor of \( p^2 \) and \( p^3 \) is \( p^2 \).
- The common factor of \( q \) and \( q^2 \) is \( q \).

So, we can factor out \( 5p^2q \):

\[
50p^2q + 35p^3q^2 = 5p^2q(10 + 7pq)
\]

### 2. \( 12a^3 - 3a^2(a + 2) \)

First, distribute the second term:

\[
12a^3 - 3a^2(a + 2) = 12a^3 - 3a^3 - 6a^2
\]

Now simplify the expression:

\[
= (12a^3 - 3a^3) - 6a^2 = 9a^3 - 6a^2
\]

The common factor is \( 3a^2 \):

\[
9a^3 - 6a^2 = 3a^2(3a - 2)
\]

### 3. \( 5(a - 3) - 2u(3 - a) \)

Let's first notice that \( 3 - a = -(a - 3) \), so we can rewrite the second term as:

\[
5(a - 3) - 2u(3 - a) = 5(a - 3) + 2u(a - 3)
\]

Now, factor out \( (a - 3) \):

\[
5(a - 3) + 2u(a - 3) = (a - 3)(5 + 2u)
\]

These are the factored forms for the given problems. Would you like further details on any of these steps or help with the remaining problems?

### Related Questions:
1. How do you factor algebraic expressions with more variables?
2. What is the difference between factoring by grouping and factoring by taking out the GCF?
3. How do you factor quadratic expressions?
4. Can all algebraic expressions be factored?
5. How do we handle expressions where the signs are reversed, like in problem 3?

### Tip:
Always look for common factors in both coefficients and variables before applying more complex factoring techniques!

Factor the following expressions: 50p^2q + 35p^3q^2, 12a^3 - 3a^2(a + 2), 5(a - 3) - 2u(3 - a)

Learn how to factor algebraic expressions like 50p^2q + 35p^3q^2, 12a^3 - 3a^2(a + 2), and 5(a - 3) - 2u(3 - a) using common factoring methods. This problem explains the greatest common factor (GCF) and the distributive property, suitable for students in Grades 9-10.

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