To solve the expression \((12x^2 + x^2 - 17x + 9x)(3x^2 - 2x - 2)\), let's simplify the first polynomial and then multiply the two expressions.

### Step 1: Simplify the first polynomial
\[
12x^2 + x^2 - 17x + 9x = 13x^2 - 8x
\]
So the expression becomes:
\[
(13x^2 - 8x)(3x^2 - 2x - 2)
\]

### Step 2: Distribute and multiply the terms
We now apply the distributive property by multiplying each term in the first polynomial by each term in the second polynomial.

\[
13x^2(3x^2 - 2x - 2) = 13x^2 \cdot 3x^2 + 13x^2 \cdot (-2x) + 13x^2 \cdot (-2)
\]
\[
= 39x^4 - 26x^3 - 26x^2
\]

Next:
\[
-8x(3x^2 - 2x - 2) = -8x \cdot 3x^2 + (-8x) \cdot (-2x) + (-8x) \cdot (-2)
\]
\[
= -24x^3 + 16x^2 + 16x
\]

### Step 3: Combine like terms
Now, let's add up all the terms:

\[
39x^4 - 26x^3 - 26x^2 - 24x^3 + 16x^2 + 16x
\]

Combine the \(x^3\) and \(x^2\) terms:
\[
39x^4 - 50x^3 - 10x^2 + 16x
\]

### Final result:
\[
39x^4 - 50x^3 - 10x^2 + 16x
\]

This is the simplified form of the given expression.

---

Would you like further clarification or additional details on any step?

Here are 5 related questions for further practice:
1. How would you multiply two polynomials with more terms?
2. Can you factor the expression \(39x^4 - 50x^3 - 10x^2 + 16x\)?
3. What happens if the polynomials had different powers of \(x\)?
4. How can the distributive property be used with three polynomials?
5. What is the result of multiplying \( (2x + 3)(x - 4) \)?

**Tip**: Always group like terms after expanding to simplify your final expression.

Learn how to multiply the polynomials (12x² + x² - 17x + 9x)(3x² - 2x - 2) step-by-step with detailed explanations and examples. This problem involves polynomial multiplication and combining like terms. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation