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Let's now solve the factorization for each expression.

1. **\( 4a^2b^3 + 6a^3b^2 \)**:
   - Factor out the greatest common factor (GCF) from both terms:
     \[
     GCF = 2a^2b^2
     \]
     Factoring out the GCF:
     \[
     4a^2b^3 + 6a^3b^2 = 2a^2b^2(2b + 3a)
     \]
   Final factored form: \( 2a^2b^2(2b + 3a) \).

2. **\( 10x^2 + x \)**:
   - The GCF of both terms is \( x \). Factoring out \( x \):
     \[
     10x^2 + x = x(10x + 1)
     \]
   Final factored form: \( x(10x + 1) \).

3. **\( 2x^2 + x - 1 \)**:
   - This is a quadratic expression, so we will factor it using trial and error (or the quadratic formula if necessary).
     We need two numbers that multiply to \( 2 \times -1 = -2 \) and add to \( 1 \). The numbers are \( 2 \) and \( -1 \). Rewriting the middle term:
     \[
     2x^2 + 2x - x - 1 = (2x(x + 1)) - 1(x + 1)
     \]
     Factoring the common term \( (x + 1) \):
     \[
     (2x - 1)(x + 1)
     \]
   Final factored form: \( (2x - 1)(x + 1) \).

4. **\( (x + 2)^2 + 3(x - 1)(x + 2) \)**:
   - Factor out \( (x + 2) \) from both terms:
     \[
     (x + 2)^2 + 3(x - 1)(x + 2) = (x + 2)((x + 2) + 3(x - 1))
     \]
     Simplifying the expression inside the parentheses:
     \[
     (x + 2) + 3(x - 1) = x + 2 + 3x - 3 = 4x - 1
     \]
     Final factored form: \( (x + 2)(4x - 1) \).

Would you like further details on any of these factorizations?

Here are some related questions:
1. How do you identify the greatest common factor when factoring?
2. What methods are available for factoring quadratic expressions?
3. How do you approach factoring trinomials where the coefficient of \(x^2\) isn't 1?
4. Can the quadratic formula always be used for factoring quadratic expressions?
5. How does factoring work when dealing with more complex polynomials?

**Tip**: Always start factoring by looking for a GCF—it simplifies the expression and makes the next steps easier!

Factor the following expressions: 4a^2b^3 + 6a^3b^2, 10x^2 + x, 2x^2 + x - 1, (x + 2)^2 + 3(x - 1)(x + 2)

Learn how to factor complex polynomials and quadratic expressions such as 4a^2b^3 + 6a^3b^2 and 2x^2 + x - 1. This guide covers essential algebra concepts, including factoring using the greatest common factor and trial and error methods. Suitable for high school students in Grades 9-10.

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