To simplify the expression $(n^2 + 6n - 4)(2n - 4)$, we can use the distributive property (also known as the FOIL method for binomials). Here's the step-by-step process:

Step 1: Expand the expression

$(n^2 + 6n - 4)(2n - 4)$

We distribute each term of the first expression $n^2 + 6n - 4$ to the second expression $2n - 4$.

1. $n^2 \cdot (2n - 4) = n^2 \cdot 2n - n^2 \cdot 4 = 2n^3 - 4n^2$
2. $6n \cdot (2n - 4) = 6n \cdot 2n - 6n \cdot 4 = 12n^2 - 24n$
3. $-4 \cdot (2n - 4) = -4 \cdot 2n - (-4 \cdot 4) = -8n + 16$

Step 2: Combine all terms

Now, sum up all the terms obtained from the distribution:

$2n^3 - 4n^2 + 12n^2 - 24n - 8n + 16$

Step 3: Simplify

Combine like terms:

• $n^2$ terms: $-4n^2 + 12n^2 = 8n^2$
• $n$ terms: $-24n - 8n = -32n$

Thus, the simplified expression is:

$2n^3 + 8n^2 - 32n + 16$

This is the fully simplified form of the expression.

Would you like further details or have any questions?

Related Questions:

1. How do you factor a cubic polynomial like $2n^3 + 8n^2 - 32n + 16$?
2. What is the distributive property, and how does it apply to polynomial multiplication?
3. Can this cubic expression be solved for its roots?
4. How does factoring differ from expanding in algebraic expressions?
5. How can polynomial identities help simplify expressions?

Tip:

When expanding binomials, keep track of each term systematically to avoid mistakes, especially when handling multiple terms in each binomial.