To find the area of the shaded region, we need to subtract the area of the inner rectangle (white part) from the area of the outer rectangle (blue part).

Step 1: Calculate the area of the outer rectangle.

• The dimensions of the outer rectangle are given as $5x - 1$ (length) and $3x + 4$ (width).
• The area of a rectangle is given by multiplying its length by its width: $\text{Area of outer rectangle} = (5x - 1)(3x + 4)$

Step 2: Calculate the area of the inner rectangle.

• The dimensions of the inner rectangle are $x + 1$ (length) and $x - 1$ (width).
• The area of the inner rectangle is: $\text{Area of inner rectangle} = (x + 1)(x - 1)$ Notice that $(x + 1)(x - 1)$ is a difference of squares, which simplifies to: $(x + 1)(x - 1) = x^2 - 1$

Step 3: Subtract the area of the inner rectangle from the area of the outer rectangle.

We need to subtract the area of the inner rectangle from the outer one to find the shaded region: $\text{Area of shaded region} = (5x - 1)(3x + 4) - (x^2 - 1)$

Step 4: Expand and simplify.

First, expand the outer rectangle's area expression: $(5x - 1)(3x + 4) = 15x^2 + 20x - 3x - 4 = 15x^2 + 17x - 4$ Now subtract the inner rectangle’s area: $15x^2 + 17x - 4 - (x^2 - 1) = 15x^2 + 17x - 4 - x^2 + 1$ Simplify further: $(15x^2 - x^2) + 17x + (-4 + 1) = 14x^2 + 17x - 3$

Final Answer:

The expression representing the area of the shaded region is: $14x^2 + 17x - 3$

Would you like more details or explanations?

Here are 5 related questions to expand your understanding:

1. How would the expression change if the dimensions of the outer rectangle were doubled?
2. Can you calculate the area of the shaded region for a specific value of $x$?
3. What happens if the inner rectangle has the same area as the outer rectangle?
4. How would the expression change if the dimensions of the inner rectangle were $x + 2$ and $x - 2$?
5. Can the area of the shaded region ever be negative? Why or why not?

Tip: When simplifying algebraic expressions involving rectangles, look for common factorizations like the difference of squares to make the process easier.