The expression shown in the image is:

$(3x^2 + 2x)(x^4 - 3x + 1)$

To simplify this, we apply the distributive property (also known as the FOIL method for binomials, but in this case, it's extended for two polynomials). This means that every term in the first polynomial $(3x^2 + 2x)$ is multiplied by every term in the second polynomial $(x^4 - 3x + 1)$.

Step-by-step solution:

Multiply $3x^2$ by each term in $x^4 - 3x + 1$:

• $3x^2 \cdot x^4 = 3x^6$
• $3x^2 \cdot (-3x) = -9x^3$
• $3x^2 \cdot 1 = 3x^2$

Multiply $2x$ by each term in $x^4 - 3x + 1$:

• $2x \cdot x^4 = 2x^5$
• $2x \cdot (-3x) = -6x^2$
• $2x \cdot 1 = 2x$

Combine the results:

Now we add all these terms together: $3x^6 + 2x^5 - 9x^3 + 3x^2 - 6x^2 + 2x$

Combine like terms:

• $3x^2 - 6x^2 = -3x^2$

Final simplified expression: $3x^6 + 2x^5 - 9x^3 - 3x^2 + 2x$

This is the expanded form of the given expression.

Would you like further details or have any questions?

Here are 5 related questions for further practice:

1. How would you expand $(2x^2 + 3)(x^3 - 4x + 2)$?
2. What is the result of multiplying $(x^2 + 5x - 1)$ by $(x^3 + 2x^2 - x)$?
3. Can you find the degree of the polynomial $2x^4 - 3x^3 + 5x - 7$?
4. How does distributing work when expanding two binomials, like $(x + 2)(x - 3)$?
5. How would the product of $(x^2 + 2x + 1)$ and $(x^2 - x + 3)$ be simplified?

Tip:

When expanding polynomials, make sure to carefully multiply each term in one polynomial by every term in the other, and then combine like terms to simplify the expression.