Multiply and simplify 



﻿open parentheses 3 x to the power of 2 space end exponent minus space 2 x space plus 8 close parentheses open parentheses negative 4 x squared space plus space 7 x space minus space 9 close parentheses﻿

To multiply and simplify the expression:

\[
(3x^2 - 2x + 8)(-4x^2 + 7x - 9)
\]

we'll distribute each term in the first parenthesis to each term in the second parenthesis.

### Step 1: Distribute \(3x^2\) to each term in \(-4x^2 + 7x - 9\):

\[
3x^2(-4x^2) + 3x^2(7x) + 3x^2(-9)
\]

This simplifies to:

\[
-12x^4 + 21x^3 - 27x^2
\]

### Step 2: Distribute \(-2x\) to each term in \(-4x^2 + 7x - 9\):

\[
-2x(-4x^2) + -2x(7x) + -2x(-9)
\]

This simplifies to:

\[
8x^3 - 14x^2 + 18x
\]

### Step 3: Distribute \(8\) to each term in \(-4x^2 + 7x - 9\):

\[
8(-4x^2) + 8(7x) + 8(-9)
\]

This simplifies to:

\[
-32x^2 + 56x - 72
\]

### Step 4: Combine all the terms:

\[
-12x^4 + 21x^3 - 27x^2 + 8x^3 - 14x^2 + 18x - 32x^2 + 56x - 72
\]

### Step 5: Combine like terms:

\[
-12x^4 + (21x^3 + 8x^3) + (-27x^2 - 14x^2 - 32x^2) + (18x + 56x) - 72
\]

Simplifies to:

\[
-12x^4 + 29x^3 - 73x^2 + 74x - 72
\]

So, the simplified expression is:

\[
\boxed{-12x^4 + 29x^3 - 73x^2 + 74x - 72}
\]

Would you like more details or have any questions?

Here are some related questions you might be interested in:
1. How do you factor a polynomial like this?
2. What are the common mistakes to avoid when distributing terms?
3. How do you multiply polynomials with more than two terms?
4. Can this polynomial expression be factored further?
5. How can you verify the correctness of the polynomial multiplication?

**Tip:** When multiplying polynomials, carefully organize and combine like terms to avoid errors.

Learn how to multiply and simplify the polynomial expression (3x^2 - 2x + 8)(-4x^2 + 7x - 9) step-by-step. This problem covers polynomial multiplication, simplification, and combining like terms. Suitable for high school students studying algebra.

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