To determine which expressions can represent the length and width of Josephine's garden, we need to find the combinations of expressions that multiply to give the area [tex]\(2x^2 + 6\)[/tex] square feet.
Let's examine the given pairs of expressions:
1. Length [tex]\( = x^2 - 3 \)[/tex] feet; Width [tex]\( = 2 \)[/tex] feet:
- Area calculation: [tex]\((x^2 - 3) \cdot 2 = 2x^2 - 6\)[/tex]
- This does not match [tex]\(2x^2 + 6\)[/tex].
2. Length [tex]\( = 2x + 3 \)[/tex] feet; Width [tex]\( = x - 2 \)[/tex] feet:
- Area calculation: [tex]\((2x + 3) \cdot (x - 2) = 2x^2 - 4x + 3x - 6 = 2x^2 - x - 6\)[/tex]
- This does not match [tex]\(2x^2 + 6\)[/tex].
3. Length [tex]\( = 2x + 2 \)[/tex] feet; Width [tex]\( = x - 3 \)[/tex] feet:
- Area calculation: [tex]\((2x + 2) \cdot (x - 3) = 2x^2 - 6x + 2x - 6 = 2x^2 - 4x - 6\)[/tex]
- This does not match [tex]\(2x^2 + 6\)[/tex].
4. Length [tex]\( = 2x - 3 \)[/tex] feet; Width [tex]\( = x + 2 \)[/tex] feet:
- Area calculation: [tex]\((2x - 3) \cdot (x + 2) = 2x^2 + 4x - 3x - 6 = 2x^2 + x - 6\)[/tex]
- This does not match [tex]\(2x^2 + 6\)[/tex].
After evaluating all given pairs, we find that none of the provided length and width expressions satisfy the given area [tex]\(2x^2 + 6\)[/tex].
Thus, none of the pairs are valid expressions for the length and width of Josephine’s garden.
