Alright, let's solve this step-by-step.
1. Determine the dimensions of the first fabric scrap:
- Width: [tex]\( 2x \)[/tex] feet
- Length: [tex]\( 5x \)[/tex] feet
2. Determine the dimensions of the entire quilt after the increases:
- The quilt will be 5 feet wider than the first scrap:
[tex]\[ \text{Width of quilt} = 2x + 5 \][/tex]
- The quilt will be 3 feet longer than the first scrap:
[tex]\[ \text{Length of quilt} = 5x + 3 \][/tex]
3. Set up the function for the area of the quilt:
- The area [tex]\( f(x) \)[/tex] of a rectangle is given by multiplying its width and length:
[tex]\[ f(x) = (\text{Width of quilt}) \times (\text{Length of quilt}) \][/tex]
- Substituting the dimensions found in step 2:
[tex]\[ f(x) = (2x + 5) \times (5x + 3) \][/tex]
4. Expand the expression:
- Use the distributive property (FOIL method) to expand:
[tex]\[ f(x) = (2x + 5)(5x + 3) \][/tex]
[tex]\[ f(x) = 2x \cdot 5x + 2x \cdot 3 + 5 \cdot 5x + 5 \cdot 3 \][/tex]
[tex]\[ f(x) = 10x^2 + 6x + 25x + 15 \][/tex]
[tex]\[ f(x) = 10x^2 + 31x + 15 \][/tex]
5. Match the result to the given options:
- The correct function corresponding to the area of the quilt, based on our expansion, is:
[tex]\[ f(x) = 10x^2 + 31x + 15 \][/tex]
Since this matches option B, the valid answer is:
Option B: [tex]\( f(x) = 10x^2 + 31x + 15 \)[/tex]
