We will expand the expression \((x^2 - 5x - 3)(x^2 + 5x - 3)\) by using the distributive property (FOIL method).

### Step-by-step expansion:

1. Expand each term in the first binomial by distributing it across the second binomial:
\[
(x^2 - 5x - 3)(x^2 + 5x - 3) = x^2(x^2 + 5x - 3) - 5x(x^2 + 5x - 3) - 3(x^2 + 5x - 3)
\]

2. Now distribute \(x^2\), \(-5x\), and \(-3\) across each term:

\[
x^2(x^2 + 5x - 3) = x^4 + 5x^3 - 3x^2
\]
\[
-5x(x^2 + 5x - 3) = -5x^3 - 25x^2 + 15x
\]
\[
-3(x^2 + 5x - 3) = -3x^2 - 15x + 9
\]

3. Combine all the terms:

\[
x^4 + 5x^3 - 3x^2 - 5x^3 - 25x^2 + 15x - 3x^2 - 15x + 9
\]

4. Simplify by combining like terms:

\[
x^4 + (5x^3 - 5x^3) + (-3x^2 - 25x^2 - 3x^2) + (15x - 15x) + 9
\]
\[
= x^4 - 31x^2 + 9
\]

### Final result:
\[
(x^2 - 5x - 3)(x^2 + 5x - 3) = x^4 - 31x^2 + 9
\]

Would you like further details on this process or have any other questions?

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Here are five questions that could help expand your understanding:

1. How can we use the distributive property (FOIL method) with polynomials in more complex scenarios?
2. What is the difference between binomial and trinomial expansion?
3. How do you factor polynomials after expanding them?
4. What other methods can be used to expand polynomials besides the distributive method?
5. How do you identify common mistakes when expanding polynomials?

**Tip**: When combining like terms, always group terms with the same degree (like \(x^3\), \(x^2\), \(x\), etc.) to simplify the expression properly.

This webpage provides a detailed solution for expanding the polynomial (x^2 – 5x – 3)(x^2 + 5x – 3) using the distributive property (FOIL method). Explore step-by-step instructions and key algebraic concepts for students in Grades 9-11.

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