To expand the expression $(2x - 7)(2x^2 + 2x - 1)$ and write it in standard polynomial form, we will use the distributive property (also known as the FOIL method in some cases) to multiply each term in the first binomial by each term in the second trinomial.

Step-by-Step Expansion:

1. Distribute $2x$ across $(2x^2 + 2x - 1)$: $2x \cdot 2x^2 = 4x^3$ $2x \cdot 2x = 4x^2$ $2x \cdot (-1) = -2x$

2. Distribute $-7$ across $(2x^2 + 2x - 1)$: $-7 \cdot 2x^2 = -14x^2$ $-7 \cdot 2x = -14x$ $-7 \cdot (-1) = 7$

Combine all the terms:

$4x^3 + 4x^2 - 2x - 14x^2 - 14x + 7$

Simplify by combining like terms:

$4x^3 + (4x^2 - 14x^2) + (-2x - 14x) + 7$ $4x^3 - 10x^2 - 16x + 7$

Thus, the expanded expression in standard form is: $4x^3 - 10x^2 - 16x + 7$

Would you like more details on any of the steps?

Here are 5 related questions to deepen your understanding:

1. How does the distributive property apply to polynomials?
2. What is the difference between expanding and factoring polynomials?
3. How can you check if your expanded polynomial is correct?
4. Can you explain the standard form of a polynomial?
5. How do you identify and combine like terms?

Tip: Always combine like terms carefully after distributing to avoid common mistakes.