A Rectangle Has A Height Of $a^2 + 3$ And A Width Of $a^2 + 2a + 5$.Express The Area Of The Entire Rectangle. Your Answer Should Be A Polynomial In Standard Form.Area $= \square$

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Introduction

In mathematics, the area of a rectangle is calculated by multiplying its height and width. Given the height of a rectangle as $a^2 + 3$ and the width as $a^2 + 2a + 5$, we are tasked with finding the area of the entire rectangle. This problem involves algebraic expressions and polynomial multiplication.

Understanding the Problem

To find the area of the rectangle, we need to multiply the height and width. The height is given as $a^2 + 3$, and the width is given as $a^2 + 2a + 5$. We will use the distributive property of multiplication to expand the expression.

Multiplying the Height and Width

To multiply the height and width, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We will apply this property to each term in the height expression and multiply it by each term in the width expression.

Multiplying the First Term in the Height by Each Term in the Width

The first term in the height is $a^2$. We will multiply this term by each term in the width:

$a^2 \cdot (a^2 + 2a + 5) = a^4 + 2a^3 + 5a^2$

Multiplying the Second Term in the Height by Each Term in the Width

The second term in the height is $3$. We will multiply this term by each term in the width:

$3 \cdot (a^2 + 2a + 5) = 3a^2 + 6a + 15$

Combining the Terms

Now, we will combine the terms we obtained in the previous steps:

$(a^4 + 2a^3 + 5a^2) + (3a^2 + 6a + 15)$

We can combine like terms by adding or subtracting the coefficients of the same degree:

$a^4 + 2a^3 + 8a^2 + 6a + 15$

Simplifying the Expression

The expression we obtained is already in standard form, so we do not need to simplify it further.

Conclusion

In this problem, we found the area of a rectangle with a height of $a^2 + 3$ and a width of $a^2 + 2a + 5$. We used the distributive property of multiplication to expand the expression and obtained the area as $a^4 + 2a^3 + 8a^2 + 6a + 15$.

Final Answer

The final answer is: $\boxed{a^4 + 2a^3 + 8a^2 + 6a + 15}$

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Introduction

In mathematics, the area of a rectangle is calculated by multiplying its height and width. Given the height of a rectangle as $a^2 + 3$ and the width as $a^2 + 2a + 5$, we are tasked with finding the area of the entire rectangle. This problem involves algebraic expressions and polynomial multiplication.

Understanding the Problem

To find the area of the rectangle, we need to multiply the height and width. The height is given as $a^2 + 3$, and the width is given as $a^2 + 2a + 5$. We will use the distributive property of multiplication to expand the expression.

Multiplying the Height and Width

To multiply the height and width, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. We will apply this property to each term in the height expression and multiply it by each term in the width expression.

Multiplying the First Term in the Height by Each Term in the Width

The first term in the height is $a^2$. We will multiply this term by each term in the width:

$a^2 \cdot (a^2 + 2a + 5) = a^4 + 2a^3 + 5a^2$

Multiplying the Second Term in the Height by Each Term in the Width

The second term in the height is $3$. We will multiply this term by each term in the width:

$3 \cdot (a^2 + 2a + 5) = 3a^2 + 6a + 15$

Combining the Terms

Now, we will combine the terms we obtained in the previous steps:

$(a^4 + 2a^3 + 5a^2) + (3a^2 + 6a + 15)$

We can combine like terms by adding or subtracting the coefficients of the same degree:

$a^4 + 2a^3 + 8a^2 + 6a + 15$

Simplifying the Expression

The expression we obtained is already in standard form, so we do not need to simplify it further.

Conclusion

In this problem, we found the area of a rectangle with a height of $a^2 + 3$ and a width of $a^2 + 2a + 5$. We used the distributive property of multiplication to expand the expression and obtained the area as $a^4 + 2a^3 + 8a^2 + 6a + 15$.

Final Answer

The final answer is: $\boxed{a^4 + 2a^3 + 8a^2 + 6a + 15}$

Q&A

Q: What is the formula for finding the area of a rectangle?

A: The formula for finding the area of a rectangle is given by the product of its height and width.

Q: How do we multiply the height and width of a rectangle?

A: We use the distributive property of multiplication to expand the expression. This involves multiplying each term in the height expression by each term in the width expression.

Q: What is the distributive property of multiplication?

A: The distributive property of multiplication states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This means that we can multiply a single term by a sum of terms by multiplying the single term by each term in the sum.

Q: How do we combine like terms in an expression?

A: We combine like terms by adding or subtracting the coefficients of the same degree. For example, if we have the expression $2x^2 + 3x^2$, we can combine the like terms by adding the coefficients: $2x^2 + 3x^2 = 5x^2$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $a^4 + 2a^3 + 8a^2 + 6a + 15$.

Q: What is the significance of this problem?

A: This problem demonstrates the use of the distributive property of multiplication to expand an expression and find the area of a rectangle. It also shows how to combine like terms in an expression.

Q: Can this problem be solved using other methods?

A: Yes, this problem can be solved using other methods, such as using the FOIL method or the box method. However, the distributive property of multiplication is a more general and powerful method for expanding expressions.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in fields such as engineering, architecture, and design. For example, finding the area of a rectangle is essential in designing buildings, bridges, and other structures.

Additional Resources

• Algebraic Expressions
• Polynomial Multiplication
• Distributive Property
• Like Terms

Conclusion

In this article, we solved a problem involving the area of a rectangle with a height of $a^2 + 3$ and a width of $a^2 + 2a + 5$. We used the distributive property of multiplication to expand the expression and obtained the area as $a^4 + 2a^3 + 8a^2 + 6a + 15$. We also answered some common questions related to the problem and provided additional resources for further learning.