factor f(x) = 5(x^2 + 3x - 2) - (2x + 4)^2 

To factor the given function \( f(x) = 5(x^2 + 3x - 2) - (2x + 4)^2 \), we start by expanding and simplifying the expression.

1. **Expand \( (2x + 4)^2 \):**

\[
(2x + 4)^2 = 4x^2 + 16x + 16
\]

2. **Substitute and simplify \( f(x) \):**

\[
f(x) = 5(x^2 + 3x - 2) - (4x^2 + 16x + 16)
\]

3. **Distribute and combine like terms:**

\[
f(x) = 5x^2 + 15x - 10 - 4x^2 - 16x - 16
\]

4. **Combine like terms:**

\[
f(x) = (5x^2 - 4x^2) + (15x - 16x) + (-10 - 16)
\]

\[
f(x) = x^2 - x - 26
\]

Now we have a quadratic function \( f(x) = x^2 - x - 26 \).

5. **Factor the quadratic expression \( x^2 - x - 26 \):**

We look for two numbers that multiply to \(-26\) and add up to \(-1\). These numbers are not readily available since \(-26\) does not have integer factors that add up to \(-1\). Therefore, this quadratic expression cannot be factored using integers.

Since the quadratic \( x^2 - x - 26 \) does not factor nicely, it can be left in its simplified form:

\[
f(x) = x^2 - x - 26
\]

Would you like more details or have any questions about the steps involved?

Here are 5 questions you might want to ask next:
1. How do you find the roots of the quadratic \( x^2 - x - 26 \)?
2. Can you explain how to complete the square for \( x^2 - x - 26 \)?
3. What are the applications of factoring in solving quadratic equations?
4. How can you graph the function \( f(x) = x^2 - x - 26 \)?
5. What are the methods to factor higher degree polynomials?

**Tip:** When factoring quadratic expressions, always check if the quadratic formula or completing the square is needed when simple factoring isn't possible.

Learn how to factor the quadratic function f(x) = 5(x^2 + 3x - 2) - (2x + 4)^2 step-by-step. This problem involves expanding and simplifying quadratic expressions, distributing terms, and combining like terms. Suitable for high school students studying quadratic equations and polynomial factoring.

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