To factor the cubic polynomial $2n^3 + 7n^2 - 2n - 7$, we can group terms and apply factoring techniques.

Step 1: Group terms

$2n^3 + 7n^2 - 2n - 7 = (2n^3 + 7n^2) - (2n + 7)$

Step 2: Factor out common terms from each group

$(2n^3 + 7n^2) - (2n + 7) = n^2(2n + 7) - 1(2n + 7)$

Step 3: Factor out the common binomial factor $(2n + 7)$

$n^2(2n + 7) - 1(2n + 7) = (n^2 - 1)(2n + 7)$

Step 4: Factor $n^2 - 1$ as a difference of squares

$n^2 - 1 = (n - 1)(n + 1)$

Step 5: Write the fully factored form

$2n^3 + 7n^2 - 2n - 7 = (n - 1)(n + 1)(2n + 7)$

Final Answer:

$\boxed{(n - 1)(n + 1)(2n + 7)}$

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Do you want more details or further clarification on the steps?

Here are five questions to expand your understanding:

1. What is the grouping method in factoring, and when is it most effective?
2. How can the difference of squares formula be applied in various contexts?
3. Could a cubic polynomial always be factored into linear and quadratic terms?
4. What techniques can be used to test if a binomial is a factor of a polynomial?
5. How does the structure of a polynomial influence your choice of factoring method?

Tip: When factoring, always check if the polynomial can first be simplified by removing a greatest common factor (GCF). This can save time in subsequent steps!