factoring solution for given problem. 10a²+35ab-45b2

To factor the expression \(10a^2 + 35ab - 45b^2\), follow these steps:

1. **Identify the greatest common factor (GCF):**
   The GCF of the coefficients \(10\), \(35\), and \(-45\) is \(5\). Factor out \(5\) from the entire expression:
   \[
   10a^2 + 35ab - 45b^2 = 5(2a^2 + 7ab - 9b^2)
   \]

2. **Factor the trinomial inside the parentheses:**
   Now, we need to factor the quadratic trinomial \(2a^2 + 7ab - 9b^2\). To do this, we look for two numbers that multiply to \(2 \times (-9) = -18\) and add to \(7\).

   These numbers are \(9\) and \(-2\), because:
   \[
   9 \times (-2) = -18 \quad \text{and} \quad 9 + (-2) = 7
   \]

3. **Rewrite the middle term:**
   Rewrite the middle term \(7ab\) using the numbers found:
   \[
   2a^2 + 9ab - 2ab - 9b^2
   \]

4. **Factor by grouping:**
   Group the terms and factor each group:
   \[
   (2a^2 + 9ab) - (2ab + 9b^2)
   \]
   Factor out the common factors from each group:
   \[
   a(2a + 9b) - b(2a + 9b)
   \]

5. **Factor out the common binomial factor:**
   Now, factor out the common binomial \(2a + 9b\):
   \[
   (a - b)(2a + 9b)
   \]

6. **Include the GCF:**
   Don't forget the GCF \(5\) that we factored out initially:
   \[
   10a^2 + 35ab - 45b^2 = 5(a - b)(2a + 9b)
   \]

Thus, the fully factored form of the expression is:
\[
\boxed{5(a - b)(2a + 9b)}
\]

Would you like more details on any step? Here are some related questions to further explore:

1. How do you determine the greatest common factor (GCF)?
2. What methods can be used to factor quadratic trinomials?
3. Why is factoring by grouping effective in this case?
4. How can you check that the factored expression is correct?
5. What are some other examples of factoring similar quadratic expressions?

**Tip:** Always double-check your factors by multiplying them back to ensure they simplify to the original expression.

Learn how to factor the expression 10a² + 35ab - 45b² with a detailed step-by-step solution. This problem covers key algebraic concepts, including factoring quadratic expressions and identifying the greatest common factor (GCF). Suitable for students in Grades 9-12.

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