https://file.aiquickdraw.com/mathbot/file/1726603672275b1q1s36t.jpg

The image shows a multiplication problem involving two binomials:

\[
(5x - 2x^2)(-5 + 3x)
\]

The table is set up to organize the terms resulting from multiplying each term in the first binomial with each term in the second binomial. Let's go step by step.

1. **First, distribute the terms**:

- Multiply \(5x\) by \(-5\)
- Multiply \(5x\) by \(3x\)
- Multiply \(-2x^2\) by \(-5\)
- Multiply \(-2x^2\) by \(3x\)

**Fill in the grid**:
- Top-left: \(5x \times -5 = -25x\)
- Top-right: \(5x \times 3x = 15x^2\)
- Bottom-left: \(-2x^2 \times -5 = 10x^2\)
- Bottom-right: \(-2x^2 \times 3x = -6x^3\)

Now, the table is complete with the following results:
\[
\begin{array}{|c|c|}
\hline
-25x & 15x^2 \\
\hline
10x^2 & -6x^3 \\
\hline
\end{array}
\]

The expanded expression combining all these terms is:

\[
-6x^3 + 15x^2 + 10x^2 - 25x
\]

Combine like terms:
\[
-6x^3 + 25x^2 - 25x
\]

That's the expanded result. Let me know if you'd like further clarification or more details!

Here are some related questions:
1. What is the general process for multiplying polynomials?
2. How do you combine like terms in polynomials?
3. What other methods can be used to multiply binomials besides the grid method?
4. How does the degree of a polynomial change when two binomials are multiplied?
5. How do negative terms affect the outcome of polynomial multiplication?

**Tip:** Always double-check signs when multiplying terms, as they can often lead to errors if overlooked.

How do you multiply the binomials (5x - 2x^2) and (-5 + 3x) using a grid method?

Learn to multiply the binomials (5x - 2x^2) and (-5 + 3x) using a grid method. This guide provides a step-by-step approach to applying the distributive property and binomial expansion to solve polynomial multiplication problems, suitable for students in Grades 8 to 10.

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