(X cube -5 xy square+1)(x square+2x-3) multiply the following

To multiply the given expressions \((x^3 - 5xy^2 + 1)\) and \((x^2 + 2x - 3)\), we'll use the distributive property, also known as the FOIL method when dealing with polynomials. However, since each term in the first polynomial needs to be multiplied by each term in the second polynomial, the process will involve distributing each term in one polynomial over all terms in the other.

### Step-by-Step Solution:

Given:
\[
(x^3 - 5xy^2 + 1)(x^2 + 2x - 3)
\]

1. **Distribute \(x^3\) over \((x^2 + 2x - 3)\):**

\[
x^3 \cdot x^2 + x^3 \cdot 2x + x^3 \cdot (-3) = x^5 + 2x^4 - 3x^3
\]

2. **Distribute \(-5xy^2\) over \((x^2 + 2x - 3)\):**

\[
-5xy^2 \cdot x^2 + (-5xy^2) \cdot 2x + (-5xy^2) \cdot (-3) = -5x^3y^2 - 10x^2y^2 + 15xy^2
\]

3. **Distribute \(1\) over \((x^2 + 2x - 3)\):**

\[
1 \cdot x^2 + 1 \cdot 2x + 1 \cdot (-3) = x^2 + 2x - 3
\]

4. **Combine all the results:**

\[
x^5 + 2x^4 - 3x^3 - 5x^3y^2 - 10x^2y^2 + 15xy^2 + x^2 + 2x - 3
\]

### Final Answer:
\[
\boxed{x^5 + 2x^4 - 3x^3 - 5x^3y^2 - 10x^2y^2 + 15xy^2 + x^2 + 2x - 3}
\]

Would you like more details on any of the steps?

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Here are some related questions:
1. How do you factorize a polynomial?
2. What is the distributive property, and how is it used in algebra?
3. How do you identify like terms in a polynomial?
4. Can you explain the FOIL method for multiplying binomials?
5. What are some common mistakes to avoid when multiplying polynomials?

**Tip:** When multiplying polynomials, carefully organize your terms to ensure you don't miss any when combining like terms.

Learn how to multiply polynomials (x^3 - 5xy^2 + 1)(x^2 + 2x - 3) with a detailed step-by-step solution using the distributive property. This problem involves polynomial multiplication and covers key algebraic concepts. Suitable for Grades 9-12.

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