To expand the equation \((x^2 - 3x - 4)^2 - 16\), follow these steps:

### Step 1: Expand the square \((x^2 - 3x - 4)^2\)
\[
(x^2 - 3x - 4)^2 = (x^2 - 3x - 4)(x^2 - 3x - 4)
\]

### Step 2: Use the distributive property (FOIL method)
Let's expand the product:

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

Now, expand each term:

- \(x^2(x^2 - 3x - 4) = x^4 - 3x^3 - 4x^2\)
- \(-3x(x^2 - 3x - 4) = -3x^3 + 9x^2 + 12x\)
- \(-4(x^2 - 3x - 4) = -4x^2 + 12x + 16\)

### Step 3: Combine all the terms
\[
x^4 - 3x^3 - 4x^2 - 3x^3 + 9x^2 + 12x - 4x^2 + 12x + 16
\]

Now, combine like terms:
\[
x^4 - 6x^3 + x^2 + 24x + 16
\]

### Step 4: Subtract 16 from the expanded polynomial
Now, substitute the expanded polynomial back into the original equation:

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

Simplify by canceling out the \(+16\) and \(-16\):

\[
x^4 - 6x^3 + x^2 + 24x
\]

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

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you expand a binomial raised to a power, like \((a + b)^2\)?
2. What are the steps to simplify polynomial expressions?
3. How do you find the roots of a polynomial equation?
4. What are some common factoring techniques for polynomials?
5. How do you apply the distributive property in algebra?

**Tip:** Always combine like terms carefully when simplifying expressions to avoid mistakes.

Learn how to expand and simplify the expression (x² - 3x - 4)² - 16 with a detailed step-by-step solution. This problem covers algebraic concepts including polynomials, the FOIL method, and binomial expansion. Suitable for students in Grades 9-12.

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