Let's go step by step to expand the expression $(2x - 7)(2x^2 + 2x - 1)$. We will use the distributive property (also known as the FOIL method for binomials) to distribute each term in the first parenthesis to every term in the second parenthesis.

Step 1: Expand using Distribution

$(2x - 7)(2x^2 + 2x - 1)$ Distribute both $2x$ and $-7$ across the second expression:

$2x(2x^2 + 2x - 1) - 7(2x^2 + 2x - 1)$

Step 2: Simplify Each Product

First, distribute $2x$: $2x \cdot 2x^2 + 2x \cdot 2x + 2x \cdot (-1) = 4x^3 + 4x^2 - 2x$

Now, distribute $-7$: $-7 \cdot 2x^2 + (-7) \cdot 2x + (-7) \cdot (-1) = -14x^2 - 14x + 7$

Step 3: Combine Like Terms

Now, add all the terms together: $4x^3 + 4x^2 - 2x - 14x^2 - 14x + 7$

Combine the $x^2$-terms and $x$-terms: $4x^3 + (4x^2 - 14x^2) + (-2x - 14x) + 7$ $4x^3 - 10x^2 - 16x + 7$

Final Answer (Polynomial in Standard Form)

$4x^3 - 10x^2 - 16x + 7$

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Do you have any questions or need further explanation?

Here are five related questions you could explore:

1. How do you factor polynomials like $4x^3 - 10x^2 - 16x + 7$?
2. How do you determine the degree of a polynomial?
3. What is the general process for distributing terms over polynomials?
4. How do you find the roots of a cubic polynomial?
5. How would the expanded polynomial change if the original expression was $(2x + 7)(2x^2 + 2x - 1)$?

Tip: When combining like terms, always group powers of $x$ systematically to avoid mistakes.