simplify (6x+2)(x^2-x-2)^2 + (3x^2 + 2x -4)(4x^3-6x^2-6x+4)

To simplify the expression \((6x + 2)(x^2 - x - 2)^2 + (3x^2 + 2x - 4)(4x^3 - 6x^2 - 6x + 4)\), we will break it into two parts and simplify each one separately.

### Step 1: Expand \((6x + 2)(x^2 - x - 2)^2\)

First, we need to square the quadratic term \((x^2 - x - 2)\).

\[
(x^2 - x - 2)^2 = (x^2 - x - 2)(x^2 - x - 2)
\]

Using the distributive property:

\[
(x^2 - x - 2)(x^2 - x - 2) = x^4 - 2x^3 - 3x^2 + 4x + 4
\]

Now, multiply this result by \((6x + 2)\):

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

Distribute \(6x\) and \(2\) across the terms:

\[
= 6x(x^4 - 2x^3 - 3x^2 + 4x + 4) + 2(x^4 - 2x^3 - 3x^2 + 4x + 4)
\]

This gives:

\[
= 6x^5 - 12x^4 - 18x^3 + 24x^2 + 24x + 2x^4 - 4x^3 - 6x^2 + 8x + 8
\]

Now, combine like terms:

\[
= 6x^5 - 10x^4 - 22x^3 + 18x^2 + 32x + 8
\]

### Step 2: Expand \((3x^2 + 2x - 4)(4x^3 - 6x^2 - 6x + 4)\)

Distribute each term from \(3x^2 + 2x - 4\) over \(4x^3 - 6x^2 - 6x + 4\):

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

First, distribute \(3x^2\):

\[
3x^2(4x^3 - 6x^2 - 6x + 4) = 12x^5 - 18x^4 - 18x^3 + 12x^2
\]

Next, distribute \(2x\):

\[
2x(4x^3 - 6x^2 - 6x + 4) = 8x^4 - 12x^3 - 12x^2 + 8x
\]

Finally, distribute \(-4\):

\[
-4(4x^3 - 6x^2 - 6x + 4) = -16x^3 + 24x^2 + 24x - 16
\]

Now, combine all the terms:

\[
12x^5 - 18x^4 - 18x^3 + 12x^2 + 8x^4 - 12x^3 - 12x^2 + 8x - 16x^3 + 24x^2 + 24x - 16
\]

Combine like terms:

\[
= 12x^5 - 10x^4 - 46x^3 + 24x^2 + 32x - 16
\]

### Step 3: Combine Both Parts

Now, add the results from Step 1 and Step 2:

\[
(6x^5 - 10x^4 - 22x^3 + 18x^2 + 32x + 8) + (12x^5 - 10x^4 - 46x^3 + 24x^2 + 32x - 16)
\]

Combine like terms:

- For \(x^5\): \(6x^5 + 12x^5 = 18x^5\)
- For \(x^4\): \(-10x^4 - 10x^4 = -20x^4\)
- For \(x^3\): \(-22x^3 - 46x^3 = -68x^3\)
- For \(x^2\): \(18x^2 + 24x^2 = 42x^2\)
- For \(x\): \(32x + 32x = 64x\)
- Constant: \(8 - 16 = -8\)

Thus, the simplified expression is:

\[
18x^5 - 20x^4 - 68x^3 + 42x^2 + 64x - 8
\]

---

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### Related Questions:
1. How do you expand a binomial squared, such as \((a + b)^2\)?
2. What are the methods to simplify polynomial expressions involving multiple terms?
3. How do you identify and combine like terms in polynomial expressions?
4. What is the distributive property and how is it applied to expand expressions?
5. What are the different techniques for factoring polynomials?

**Tip**: Always double-check your work when combining like terms to avoid missing any terms or making sign errors!

Learn how to simplify complex polynomial expressions like (6x + 2)(x^2 - x - 2)^2 + (3x^2 + 2x - 4)(4x^3 - 6x^2 - 6x + 4) with step-by-step guidance on expanding polynomials and combining like terms. Suitable for high school algebra students.

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