you are planting strawberries in your rectangular garden and need to buy soil. it has a width of 4x+8 and length of 2x^2-x+3. what is the total area of his garden

To find the total area of the rectangular garden, you multiply the width by the length. Given the width \( 4x + 8 \) and the length \( 2x^2 - x + 3 \), the area \( A \) of the garden can be calculated as:

\[
A = \text{width} \times \text{length}
\]

\[
A = (4x + 8) \times (2x^2 - x + 3)
\]

Now, distribute each term in the width expression to each term in the length expression:

\[
A = 4x \cdot (2x^2 - x + 3) + 8 \cdot (2x^2 - x + 3)
\]

Let's calculate each product:

1. \( 4x \cdot 2x^2 = 8x^3 \)
2. \( 4x \cdot (-x) = -4x^2 \)
3. \( 4x \cdot 3 = 12x \)

Next:

1. \( 8 \cdot 2x^2 = 16x^2 \)
2. \( 8 \cdot (-x) = -8x \)
3. \( 8 \cdot 3 = 24 \)

Now, combine all the terms:

\[
A = 8x^3 - 4x^2 + 12x + 16x^2 - 8x + 24
\]

Combine like terms:

\[
A = 8x^3 + 12x^2 + 4x + 24
\]

So, the total area of the garden is:

\[
\boxed{8x^3 + 12x^2 + 4x + 24}
\]

Would you like further clarification or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you factor the polynomial \( 8x^3 + 12x^2 + 4x + 24 \)?
2. How would the area change if the width or length increased by a constant amount?
3. How do you calculate the perimeter of this rectangular garden?
4. What is the significance of each term in the polynomial \( 8x^3 + 12x^2 + 4x + 24 \)?
5. How do you determine the critical points for the function representing the area?

**Tip:** When multiplying polynomials, make sure to distribute each term in one polynomial to every term in the other polynomial. This helps avoid missing any terms in the final expression.

Learn how to calculate the total area of a rectangular garden using algebraic expressions. Understand the multiplication of polynomials to find the area, with a step-by-step solution provided. Suitable for Grades 9-12.

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