Factor 6x ^ 3 * y ^ 3 + 45x ^ 2 * y ^ 2 + 21xy

Example 2: Factor x(b ^ 2 - a) + 6(b ^ 2 - a)

Let's factor each expression step-by-step.

### 1. Factoring \( 6x^3 y^3 + 45x^2 y^2 + 21xy \)

This is a cubic expression in terms of \( x \) and \( y \), but we can first factor out the greatest common factor (GCF) from all the terms.

- The common factors among the terms are \( 3xy \). Factoring \( 3xy \) from each term:

\[
6x^3 y^3 + 45x^2 y^2 + 21xy = 3xy(2x^2 y^2 + 15xy + 7)
\]

Now, let's factor the quadratic expression inside the parentheses: \( 2x^2 y^2 + 15xy + 7 \).

We'll use the **trial and error** method or **factoring by grouping**:
- The factors of \( 2x^2 y^2 \) are \( (2xy) \) and \( (xy) \).
- The factors of \( 7 \) are \( (7) \) and \( (1) \).

Thus, the quadratic can be factored as:
\[
2x^2 y^2 + 15xy + 7 = (2xy + 7)(xy + 1)
\]

So, the complete factorization of the expression is:
\[
3xy(2xy + 7)(xy + 1)
\]

### 2. Factoring \( x(b^2 - a) + 6(b^2 - a) \)

This expression has a common factor, which is \( (b^2 - a) \). We can factor it out:

\[
x(b^2 - a) + 6(b^2 - a) = (b^2 - a)(x + 6)
\]

### Final Answers:
1. \( 6x^3 y^3 + 45x^2 y^2 + 21xy = 3xy(2xy + 7)(xy + 1) \)
2. \( x(b^2 - a) + 6(b^2 - a) = (b^2 - a)(x + 6) \)

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Would you like more details on any of the steps or concepts used?

Here are 5 related questions for further practice:

1. Factor \( 8x^3y^3 + 64x^2y^2 + 32xy \).
2. Factor \( 12x^4y^4 + 36x^3y^3 + 24x^2y^2 \).
3. Factor \( x(a^2 - b) + 9(a^2 - b) \).
4. Factor \( 10x(b^3 - c) + 5(b^3 - c) \).
5. Factor \( 7x^3y + 14x^2y^2 + 21xy^3 \).

**Tip**: Always check for the greatest common factor (GCF) first when factoring any expression—it simplifies the process significantly!

Factor 6x^3 * y^3 + 45x^2 * y^2 + 21xy. Example 2: Factor x(b^2 - a) + 6(b^2 - a)

Learn how to factor expressions like 6x^3y^3 + 45x^2y^2 + 21xy and x(b^2 - a) + 6(b^2 - a) by using the greatest common factor and factoring by grouping. This detailed step-by-step solution explains key algebraic concepts and techniques, suitable for students in Grades 9-11.

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