The image you uploaded contains two different problems.

Problem 1:

The first image shows a rectangle with a right triangle cut out. The full rectangle has dimensions of 12 inches by 8 inches, and the area of the shaded region (which is the remaining part of the rectangle after the triangle is cut out) is 78 square inches. The task is to find the value of $x$, which is the length of the side of the right triangle.

To find $x$, we can use the area formula for a triangle, which is:

$\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height}$

The area of the full rectangle is:

$\text{Area of Rectangle} = 12 \times 8 = 96 \, \text{square inches}$

The area of the triangle cut out is the difference between the area of the rectangle and the shaded area:

$\text{Area of Triangle} = 96 - 78 = 18 \, \text{square inches}$

Now, applying the area formula for the triangle:

$\frac{1}{2} \times 12 \times x = 18$

Solving for $x$:

$6 \times x = 18 \quad \Rightarrow \quad x = 3$

So, $x = 3$ inches.

Problem 2:

The second problem is about calculating how many bags of grass seed are needed to cover a lawn area. The lawn is in a rectangular shape, with dimensions given in yards: 90 yards by 18 yards. The patio is located inside the lawn, with dimensions of 54 yards by 10 yards. One bag of grass seed covers an area of 900 square yards.

We first calculate the area of the lawn and the patio:

$\text{Area of Lawn} = 90 \times 18 = 1620 \, \text{square yards}$

$\text{Area of Patio} = 54 \times 10 = 540 \, \text{square yards}$

The area of the lawn that needs to be seeded is the area of the lawn minus the area of the patio:

$\text{Area to Seed} = 1620 - 540 = 1080 \, \text{square yards}$

Now, we calculate the number of bags needed:

$\text{Number of Bags} = \frac{1080}{900} = 1.2$

Since you can't buy a fraction of a bag, you will need 2 bags of seed.

Would you like further explanations or have any other questions? Here are some related questions to consider:

1. What would happen if the dimensions of the rectangle in problem 1 were different?
2. How can the area of the triangle be calculated if the base and height are unknown?
3. What other methods could be used to calculate the number of bags of grass seed?
4. How would you adjust the calculation if the patio area was a different shape?
5. How can we calculate the area of irregular shapes by dividing them into rectangles and triangles?

Tip: When solving area problems, always start by calculating the total area and then subtract any areas that are not part of the region you're interested in.