https://file.aiquickdraw.com/mathbot/file/1728239013745q0rul7an.jpg

The polynomial given in the image is:

\[
a^2 + 6ab - 40b^2
\]

Let's factor it step-by-step.

### Step 1: Multiply the coefficient of \(a^2\) (which is 1) by the constant term \(-40b^2\).
\[
1 \times (-40b^2) = -40b^2
\]

### Step 2: Find two numbers that multiply to \(-40b^2\) and add to \(6b\) (the coefficient of the middle term).
The two numbers are \(10b\) and \(-4b\), because:
\[
10b \times (-4b) = -40b^2
\]
\[
10b + (-4b) = 6b
\]

### Step 3: Rewrite the middle term \(6ab\) as \(10ab - 4ab\):
\[
a^2 + 10ab - 4ab - 40b^2
\]

### Step 4: Group the terms:
\[
(a^2 + 10ab) - (4ab + 40b^2)
\]

### Step 5: Factor each group:
\[
a(a + 10b) - 4b(a + 10b)
\]

### Step 6: Factor out the common binomial factor \((a + 10b)\):
\[
(a - 4b)(a + 10b)
\]

So, the factored form of the polynomial is:

\[
(a - 4b)(a + 10b)
\]

Let me know if you would like further clarification or have any questions!

### Related Questions:
1. How can you verify if the factorization of a polynomial is correct?
2. What other methods can be used to factor trinomials?
3. Can this polynomial be factored using the quadratic formula?
4. What happens if the middle term has a different coefficient? How does it affect factorization?
5. Is it possible to factor polynomials that don't have rational roots?

### Tip:
When factoring, always look for a common factor among the terms before trying to break it down further!

Learn how to factor the polynomial a^2 + 6ab - 40b^2 using the grouping method. This problem covers key algebraic concepts and techniques for factoring trinomials. Suitable for students in Grades 8-10.

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