Let's solve this problem step by step:

Step 1: Area of the Rectangle

The formula for the area of a rectangle is: $\text{Area of rectangle} = \text{length} \times \text{width}$ For this rectangle, the length is $(5 + 2x)$ inches and the width is 10 inches. So, the area of the rectangle is: $\text{Area of rectangle} = (5 + 2x) \times 10 = 10(5 + 2x) = 50 + 20x \, \text{square inches}$

Step 2: Area of the Triangle

The formula for the area of a triangle is: $\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ For this triangle, the base is 30 inches and the height is $(4x - 10)$ inches. So, the area of the triangle is: $\text{Area of triangle} = \frac{1}{2} \times 30 \times (4x - 10) = 15 \times (4x - 10) = 60x - 150 \, \text{square inches}$

Step 3: Set the Areas Equal

Since the area of the rectangle is equal to the area of the triangle, we can set the two expressions equal to each other: $50 + 20x = 60x - 150$

Step 4: Solve for $x$

Now, let's solve the equation: $50 + 20x = 60x - 150$ Subtract $20x$ from both sides: $50 = 40x - 150$ Add 150 to both sides: $200 = 40x$ Divide by 40: $x = 5$

Step 5: Conclusion

The value of $x$ is 5.

Would you like further details or clarifications?

Here are some related questions to explore:

1. What would happen if the width of the rectangle was doubled?
2. How would the areas change if the height of the triangle increased by 5 inches?
3. What is the perimeter of the rectangle when $x = 5$?
4. How do the areas of the two shapes compare when $x = 0$?
5. Can you derive a general formula for $x$ based on different lengths of the rectangle?

Tip: In geometry problems, always double-check the units to ensure consistency in your calculations.