Let’s analyze and solve the problem from the uploaded image.

Given:

1. Rectangular Home:

   • Dimensions: $(3 + 10x) \, \text{yd}$ and $(2 + 5x) \, \text{yd}$
   • Perimeter formula for a rectangle: $P = 2(\text{length} + \text{width})$
   = 2(5 + 15x) = 10 + 30x \, \text{yd}.$$

2. Square Pool:

   • Side length: $2x \, \text{yd}$
   • Perimeter formula for a square: $P = 4 \times \text{side}$

   $P_{\text{pool}} = 4(2x) = 8x \, \text{yd}.$

3. Total Fencing Needed:

   • Combined perimeter: $P_{\text{total}} = P_{\text{home}} + P_{\text{pool}} + 10$
   • Substituting:
   = 20 + 38x \, \text{yd}.$$

Solution with $x = 5$:

1. Home's Perimeter: $P_{\text{home}} = 10 + 30(5) = 10 + 150 = 160 \, \text{yd}.$

2. Pool's Perimeter: $P_{\text{pool}} = 8(5) = 40 \, \text{yd}.$

Final Results:

• Total amount of fencing expression: $20 + 38x$
• Home's fencing needed: $160 \, \text{yd}$
• Pool's fencing needed: $40 \, \text{yd}$

Questions to Explore Further:

1. What is the formula for the area of the home and pool?
2. How does the total fencing change with a different $x$ value?
3. If the pool's side length increases, how will it affect the fencing needed?
4. Can the perimeter expressions be simplified further using different units?
5. What is the cost of fencing if it’s priced per yard?

Tip:

When working with variable expressions, double-check by substituting specific values to verify correctness!