A rectangle has a height of 
\[7a^2\] and a width of 
\[a^4+5a^2+4\].
Express the area of the entire rectangle.
Your answer should be a polynomial in standard form.
A rectangle with a height of 7 A squared and a width of A to the fourth power plus 5 A squared plus 4. The rectangle has 3 sections, one for each term of the width.


To find the area of the rectangle, we need to multiply the height and the width together. The area \( A \) of the rectangle is given by:

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

Given:
- Height = \( 7a^2 \)
- Width = \( a^4 + 5a^2 + 4 \)

The area is:

\[
A = 7a^2 \times (a^4 + 5a^2 + 4)
\]

Now, to calculate this, distribute \( 7a^2 \) to each term in the width expression:

\[
A = 7a^2 \times a^4 + 7a^2 \times 5a^2 + 7a^2 \times 4
\]

Simplifying each term:

\[
A = 7a^6 + 35a^4 + 28a^2
\]

Thus, the area of the rectangle is:

\[
A = 7a^6 + 35a^4 + 28a^2
\]

This is the area expressed as a polynomial in standard form.

Would you like more details on the solution or have any questions? 

Here are some related questions:
1. How do we simplify polynomials involving multiple terms?
2. What are the properties of a polynomial in standard form?
3. How can we interpret the area of a rectangle in terms of algebraic expressions?
4. What happens if we change the height or width of the rectangle?
5. How can the distributive property be used in polynomial multiplication?

Tip: Always ensure you correctly distribute terms when multiplying polynomials to avoid missing any components!

A rectangle has a height of 7a^2 and a width of a^4+5a^2+4. Express the area of the entire rectangle. Your answer should be a polynomial in standard form. A rectangle with a height of 7 A squared and a width of A to the fourth power plus 5 A squared plus 4. The rectangle has 3 sections, one for each term of the width.

Learn how to calculate the area of a rectangle when its dimensions are expressed as polynomials. This solution involves multiplying a polynomial height by a polynomial width, simplifying the result, and expressing it in standard form. Suitable for students in Grades 7-9.

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