To simplify the expression \((4x + 7)(x^2 + x - 6)\), we will expand the terms using the distributive property (also known as FOIL for binomials). Here's the step-by-step breakdown:

\[
(4x + 7)(x^2 + x - 6)
\]

We will multiply each term in \((4x + 7)\) by each term in \((x^2 + x - 6)\):

1. \(4x \times x^2 = 4x^3\)
2. \(4x \times x = 4x^2\)
3. \(4x \times -6 = -24x\)
4. \(7 \times x^2 = 7x^2\)
5. \(7 \times x = 7x\)
6. \(7 \times -6 = -42\)

Now, add all the terms together:

\[
4x^3 + 4x^2 - 24x + 7x^2 + 7x - 42
\]

Next, combine like terms:

- \(4x^2 + 7x^2 = 11x^2\)
- \(-24x + 7x = -17x\)

Thus, the simplified expression is:

\[
4x^3 + 11x^2 - 17x - 42
\]

Would you like any further explanation?

### Related Questions:
1. How do you factor cubic polynomials?
2. What is the distributive property in algebra?
3. Can this expression be factored further?
4. How do you solve for roots in a cubic equation?
5. What are the applications of polynomial expansion?

### Tip:
When expanding polynomials, always keep track of like terms to simplify the final expression easily.

Learn how to expand and simplify the polynomial expression (4x + 7)(x² + x - 6) step by step using the distributive property. This example demonstrates key algebraic concepts such as polynomial expansion, combining like terms, and applying the distributive property. Suitable for students in Grades 9-12.

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