To determine the expressions that can represent the length and width of Josephine's rectangular garden with an area of [tex]\( 2x^2 + x - 6 \)[/tex] square feet, we need to factor the quadratic expression [tex]\( 2x^2 + x - 6 \)[/tex].
Step-by-Step Solution:
1. Identify the quadratic expression: The given expression for the area of the garden is [tex]\( 2x^2 + x - 6 \)[/tex].
2. Factor the quadratic expression: We need to find factors of [tex]\( 2x^2 + x - 6 \)[/tex] that represent the possible length and width of the garden.
3. Write the expression as a product of two binomials: We are looking for two binomials whose product will give us the quadratic expression. Factoring [tex]\( 2x^2 + x - 6 \)[/tex], we get:
[tex]\( (x + 2)(2x - 3) \)[/tex]
This implies the factored form of the quadratic equation is [tex]\( (x + 2)(2x - 3) \)[/tex].
4. Determine the possible dimensions (length and width): From the factored form, the expressions for the length and width can be directly read off as the factors:
- One factor is [tex]\( x + 2 \)[/tex]
- The other factor is [tex]\( 2x - 3 \)[/tex]
5. Match the factors to the given options:
- Option 1: length [tex]\( = x^2 - 3 \)[/tex] feet; width [tex]\( = 2 \)[/tex] feet
- This option is incorrect because neither factor matches [tex]\( x^2 - 3 \)[/tex].
- Option 2: length [tex]\( = 2x + 3 \)[/tex] feet; width [tex]\( = x - 2 \)[/tex] feet
- This option is incorrect because neither factor [tex]\( 2x + 3 \)[/tex] nor [tex]\( x - 2 \)[/tex] matches our factors of [tex]\( x + 2 \)[/tex] or [tex]\( 2x - 3 \)[/tex].
- Option 3: length [tex]\( = 2x + 2 \)[/tex] feet; width [tex]\( = x - 3 \)[/tex] feet
- This option is incorrect because neither factor [tex]\( 2x + 2 \)[/tex] nor [tex]\( x - 3 \)[/tex] matches our factors of [tex]\( x + 2 \)[/tex] or [tex]\( 2x - 3 \)[/tex].
- Option 4: length [tex]\( = 2x - 3 \)[/tex] feet; width [tex]\( = x + 2 \)[/tex] feet
- This option is correct since both factors [tex]\( 2x - 3 \)[/tex] and [tex]\( x + 2 \)[/tex] match the factors obtained.
Therefore, the expressions that can represent the length and width of Josephine's garden are:
[tex]\[ \text{Option 4: length} = 2x - 3 \text{ feet; width} = x + 2 \text{ feet.} \][/tex]
