To expand the expression \((3u^2 - 2u - 4)(2u^2 + 7u + 6)\), we will use the distributive property (also known as the FOIL method for binomials). This involves multiplying each term in the first polynomial by each term in the second polynomial.

Let's break it down:

\[
(3u^2 - 2u - 4)(2u^2 + 7u + 6)
\]

1. **First, distribute \(3u^2\) across the second polynomial:**

\[
3u^2 \times 2u^2 = 6u^4
\]
\[
3u^2 \times 7u = 21u^3
\]
\[
3u^2 \times 6 = 18u^2
\]

2. **Next, distribute \(-2u\) across the second polynomial:**

\[
-2u \times 2u^2 = -4u^3
\]
\[
-2u \times 7u = -14u^2
\]
\[
-2u \times 6 = -12u
\]

3. **Finally, distribute \(-4\) across the second polynomial:**

\[
-4 \times 2u^2 = -8u^2
\]
\[
-4 \times 7u = -28u
\]
\[
-4 \times 6 = -24
\]

Now, combine all these terms together:

\[
6u^4 + 21u^3 + 18u^2 - 4u^3 - 14u^2 - 12u - 8u^2 - 28u - 24
\]

Combine like terms:

\[
6u^4 + (21u^3 - 4u^3) + (18u^2 - 14u^2 - 8u^2) + (-12u - 28u) - 24
\]

Simplify:

\[
6u^4 + 17u^3 - 4u^2 - 40u - 24
\]

So, the expanded form of \((3u^2 - 2u - 4)(2u^2 + 7u + 6)\) is:

\[
\boxed{6u^4 + 17u^3 - 4u^2 - 40u - 24}
\]

Would you like more details or have any questions?

### Related Questions:
1. How can you factor the expression \(6u^4 + 17u^3 - 4u^2 - 40u - 24\)?
2. What are the roots of the polynomial \(6u^4 + 17u^3 - 4u^2 - 40u - 24\)?
3. How would the expansion change if the coefficients in the polynomials were different?
4. How do you apply the distributive property in higher degree polynomials?
5. Can you graph the polynomial \(6u^4 + 17u^3 - 4u^2 - 40u - 24\) and discuss its key features?

### Tip:
When expanding polynomials, always carefully distribute each term to avoid common mistakes, such as missing a term or incorrect signs.

Learn how to expand the expression (3u^2 - 2u - 4)(2u^2 + 7u + 6) using the distributive property (FOIL method). This detailed solution covers each step of the expansion process, including combining like terms. Suitable for students in Grades 8-10.

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