Question

1. tim wants to have a rectangular garden with a walkway around it. what is the area of the garden with the walkway? image of rectangles 6. simplify the polynomial. \\((5 - 2m^4 + 2m^3) - (2m^2 + 4m^4 - 1)\\) solution: 7. you are designing a rectangular garden with a smaller rectangular flower bed inside it. - the garden length is \\(3x + 5\\) meters and the width is \\(2x + 4\\) meters. - the flower bed inside the garden has a length of \\(x + 2\\) meters and a width of \\(x + 1\\) meters. determine the area of the garden that does not include the flowerbed. area: 8. how much greater is the area of the rectangle, figure a than the square, figure b? image of figure a (rectangle) and figure b (square) area of rectangle a is greater than square b by:

Explanation:

Problem 6: Simplify the polynomial \((5 - 2m^4 + 2m^3) - (2m^2 + 4m^4 - 1)\)

Step 1: Distribute the negative sign

We need to distribute the negative sign to each term inside the second parentheses. This gives us:
\(5 - 2m^4 + 2m^3 - 2m^2 - 4m^4 + 1\)

Step 2: Combine like terms

• For the \(m^4\) terms: \(-2m^4 - 4m^4 = -6m^4\)
• The \(m^3\) term remains as \(2m^3\) (since there are no other \(m^3\) terms to combine with)
• The \(m^2\) term remains as \(-2m^2\) (since there are no other \(m^2\) terms to combine with)
• For the constant terms: \(5 + 1 = 6\)

Putting it all together, we get:
\(-6m^4 + 2m^3 - 2m^2 + 6\)

Step 1: Find the area of the garden

The garden is a rectangle with length \(3x + 5\) and width \(2x + 4\). The area of a rectangle is given by \(A = \text{length} \times \text{width}\). So, the area of the garden is:

$$
LATEXBLOCK0
$$

Step 2: Find the area of the flowerbed

The flowerbed is a rectangle with length \(x + 2\) and width \(x + 1\). Using the area formula for a rectangle, the area of the flowerbed is:

$$
LATEXBLOCK1
$$

Step 3: Subtract the area of the flowerbed from the area of the garden

To find the area of the garden excluding the flowerbed, we subtract the area of the flowerbed from the area of the garden:

$$
LATEXBLOCK2
$$

Step 1: Find the area of figure A (rectangle)

Figure A is a rectangle with length \(3x + 2\) and width \(x + 3\). The area of a rectangle is \(\text{length} \times \text{width}\), so:

$$
LATEXBLOCK0
$$

Step 2: Find the area of figure B (square)

Figure B is a square with side length \(x + 4\). The area of a square is \(\text{side length}^2\), so:

$$
LATEXBLOCK1
$$

Step 3: Find the difference in areas

To find how much greater the area of figure A is than figure B, we subtract the area of figure B from the area of figure A:

$$
LATEXBLOCK2
$$

Answer:

\(-6m^4 + 2m^3 - 2m^2 + 6\)

Problem 7: Determine the area of the garden that does not include the flowerbed