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We are tasked with multiplying two polynomials:

\[
(6x^2 - 5x + 3) \times (3x^2 + 7x - 3)
\]

### Step-by-Step Solution:

1. **Distribute each term in the first polynomial** to every term in the second polynomial.

\[
(6x^2 - 5x + 3) \times (3x^2 + 7x - 3)
\]

We break this down into multiple distributions.

#### Distribute \(6x^2\) to each term in \((3x^2 + 7x - 3)\):

\[
6x^2 \times 3x^2 = 18x^4
\]
\[
6x^2 \times 7x = 42x^3
\]
\[
6x^2 \times -3 = -18x^2
\]

#### Distribute \(-5x\) to each term in \((3x^2 + 7x - 3)\):

\[
-5x \times 3x^2 = -15x^3
\]
\[
-5x \times 7x = -35x^2
\]
\[
-5x \times -3 = 15x
\]

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

\[
3 \times 3x^2 = 9x^2
\]
\[
3 \times 7x = 21x
\]
\[
3 \times -3 = -9
\]

### Combine all the terms:

\[
18x^4 + 42x^3 - 18x^2 - 15x^3 - 35x^2 + 15x + 9x^2 + 21x - 9
\]

Now, group like terms:

\[
18x^4 + (42x^3 - 15x^3) + (-18x^2 - 35x^2 + 9x^2) + (15x + 21x) - 9
\]

Simplify:

\[
18x^4 + 27x^3 - 44x^2 + 36x - 9
\]

### Final Answer:

\[
18x^4 + 27x^3 - 44x^2 + 36x - 9
\]

Would you like further clarification or additional steps explained?

Here are five related questions to extend your learning:
1. How do you factor polynomials like \(18x^4 + 27x^3 - 44x^2 + 36x - 9\)?
2. Can you verify this polynomial by expanding it using a different method, like synthetic division?
3. What are the roots of this polynomial, and how do you find them?
4. How does the degree of the polynomial affect its graph?
5. How would the solution change if one of the polynomials had a higher degree?

**Tip**: When multiplying polynomials, always keep track of like terms to simplify the final expression efficiently.

Multiply (6x^2 - 5x + 3) by (3x^2 + 7x - 3)

Learn how to multiply the polynomials (6x^2 - 5x + 3) by (3x^2 + 7x - 3) with a detailed step-by-step solution. This problem covers key algebraic concepts, including the distributive property for polynomials. Suitable for high school students.

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