To determine how much wallpaper is required, we need to subtract the areas of the window and the shelving unit from the total area of the wall. Here's the breakdown:

Step 1: Total Area of the Wall

The area of the wall is given as:

$\text{Area of wall} = 3x^2 + 4x + 1 \quad \text{(square units)}$

Step 2: Area of the Window

The window is a rectangle, and its area is given by multiplying the width and length:

$\text{Area of window} = (x + 2)(x + 1)$

To find this area, we expand the expression:

$(x + 2)(x + 1) = x^2 + x + 2x + 2 = x^2 + 3x + 2$

So, the area of the window is:

$\text{Area of window} = x^2 + 3x + 2 \quad \text{(square units)}$

Step 3: Area of the Shelving Unit

The area of the shelving unit is given as:

$\text{Area of shelving unit} = x^2 + 5x + 6 \quad \text{(square units)}$

Step 4: Area to be Covered with Wallpaper

To find the area that needs to be covered with wallpaper, subtract the area of the window and the area of the shelving unit from the total wall area:

$\text{Area to cover} = (3x^2 + 4x + 1) - (x^2 + 3x + 2) - (x^2 + 5x + 6)$

Now, let's simplify this expression:

$= 3x^2 + 4x + 1 - x^2 - 3x - 2 - x^2 - 5x - 6$ $= (3x^2 - x^2 - x^2) + (4x - 3x - 5x) + (1 - 2 - 6)$ $= x^2 - 4x - 7$

Final Answer:

The area that needs to be covered with wallpaper is:

$\boxed{x^2 - 4x - 7} \quad \text{(square units)}$

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Do you need further details or clarification on any step? Here are some related questions you might find interesting:

1. How would the total area change if the dimensions of the window were different?
2. What if the shelving unit had a different shape, such as a circular or triangular one?
3. How do we factor the quadratic expression $x^2 - 4x - 7$?
4. How would the wallpaper requirement change if the wall area were to increase by a factor of 2?
5. What is the relationship between the size of the window and the required wallpaper area?

Tip: When simplifying expressions involving multiple terms, be sure to group like terms (like the $x^2$'s, $x$'s, and constant terms) before combining them.