To determine the possible length and width expressions that represent the area [tex]\(10x^2 - 13x - 14\)[/tex] of Mandisa's yard, we need to factorize the given quadratic expression.
The quadratic expression is:
[tex]\[10x^2 - 13x - 14\][/tex]
Through factorization, we express this quadratic function as a product of two binomials. The result of this factorization is:
[tex]\[(x - 2)(10x + 7)\][/tex]
Comparing the factored form [tex]\((x - 2)(10x + 7)\)[/tex] with the potential answers given:
1. [tex]\(2x + 2\)[/tex] and [tex]\(5x - 2\)[/tex]
2. [tex]\(5x - 2\)[/tex] and [tex]\(2x + 7\)[/tex]
3. [tex]\(x - 2\)[/tex] and [tex]\(10x + 7\)[/tex]
4. [tex]\(10x + 7\)[/tex] and [tex]\(x - 2\)[/tex]
We see that the correct expressions that correspond to the area [tex]\(10x^2 - 13x - 14\)[/tex] are:
[tex]\[\boxed{(x - 2) \text{ and } (10x + 7)}\][/tex]
Thus, the expressions that represent the length and width of the rectangle are:
[tex]\[(x - 2)\][/tex]
and
[tex]\[(10x + 7)\][/tex]
Alternatively, since the multiplication is commutative, the expressions could also be:
[tex]\[(10x + 7)\][/tex]
and
[tex]\[(x - 2)\][/tex]
These are the valid expressions for the length and width of Mandisa's yard, given her specified area.
