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!