if m(x)=2x^2+5x+2 and n(x)=4x^2-3x+1, find m(x) times n(x)

To find \(m(x) \times n(x)\) where \(m(x) = 2x^2 + 5x + 2\) and \(n(x) = 4x^2 - 3x + 1\), we need to multiply the two polynomials together. 

Given:
\[ m(x) = 2x^2 + 5x + 2 \]
\[ n(x) = 4x^2 - 3x + 1 \]

We will multiply each term of \(m(x)\) by each term of \(n(x)\):

\[
(2x^2 + 5x + 2)(4x^2 - 3x + 1)
\]

### Step-by-step multiplication:

1. Multiply \(2x^2\) by each term in \(n(x)\):
   \[
   2x^2 \cdot 4x^2 = 8x^4
   \]
   \[
   2x^2 \cdot (-3x) = -6x^3
   \]
   \[
   2x^2 \cdot 1 = 2x^2
   \]

2. Multiply \(5x\) by each term in \(n(x)\):
   \[
   5x \cdot 4x^2 = 20x^3
   \]
   \[
   5x \cdot (-3x) = -15x^2
   \]
   \[
   5x \cdot 1 = 5x
   \]

3. Multiply \(2\) by each term in \(n(x)\):
   \[
   2 \cdot 4x^2 = 8x^2
   \]
   \[
   2 \cdot (-3x) = -6x
   \]
   \[
   2 \cdot 1 = 2
   \]

### Combine all terms:

\[
8x^4 + (-6x^3) + 2x^2 + 20x^3 + (-15x^2) + 5x + 8x^2 + (-6x) + 2
\]

### Simplify by combining like terms:

\[
8x^4 + (-6x^3 + 20x^3) + (2x^2 - 15x^2 + 8x^2) + (5x - 6x) + 2
\]

\[
8x^4 + 14x^3 - 5x^2 - x + 2
\]

So, the product \(m(x) \times n(x)\) is:

\[
\boxed{8x^4 + 14x^3 - 5x^2 - x + 2}
\]

Would you like details or have any questions about this solution?

Here are some related questions:

1. What is the degree of the polynomial \(8x^4 + 14x^3 - 5x^2 - x + 2\)?
2. How do you find the roots of a polynomial?
3. Can you differentiate the product \(m(x) \times n(x)\)?
4. How would you find the value of \(m(x) \times n(x)\) at \(x = 1\)?
5. What are the coefficients of each term in the polynomial \(8x^4 + 14x^3 - 5x^2 - x + 2\)?
6. How do you multiply two polynomials using the distributive property?
7. What is the significance of the leading coefficient in a polynomial?
8. Can the polynomial \(8x^4 + 14x^3 - 5x^2 - x + 2\) be factored further?

**Tip:** When multiplying polynomials, it is helpful to align terms by their degrees to ensure all like terms are correctly combined.

Learn how to multiply polynomials m(x) = 2x^2 + 5x + 2 and n(x) = 4x^2 - 3x + 1 step-by-step. Understand the process of combining like terms and the distributive property in polynomial multiplication. Suitable for high school students.

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