To determine which choice is equivalent to the expression [tex]\((\sqrt{x} + 2)(\sqrt{x} - 3)\)[/tex], we will use the FOIL method. The FOIL method stands for First, Outer, Inner, and Last, which are the terms that we need to multiply in a binomial product.
1. First: Multiply the first terms in each binomial.
[tex]\[ \sqrt{x} \cdot \sqrt{x} = x \][/tex]
2. Outer: Multiply the outer terms in each binomial.
[tex]\[ \sqrt{x} \cdot (-3) = -3\sqrt{x} \][/tex]
3. Inner: Multiply the inner terms in each binomial.
[tex]\[ 2 \cdot \sqrt{x} = 2\sqrt{x} \][/tex]
4. Last: Multiply the last terms in each binomial.
[tex]\[ 2 \cdot (-3) = -6 \][/tex]
Now, we add all these products together:
[tex]\[ x + (-3\sqrt{x}) + (2\sqrt{x}) + (-6) \][/tex]
Combine the like terms:
[tex]\[ x - 3\sqrt{x} + 2\sqrt{x} - 6 \][/tex]
[tex]\[ x - \sqrt{x} - 6 \][/tex]
So, the expression [tex]\((\sqrt{x} + 2)(\sqrt{x} - 3)\)[/tex] simplifies to:
[tex]\[ x - \sqrt{x} - 6 \][/tex]
Therefore, the correct choice is:
[tex]\[ \text{D. } x - \sqrt{x} - 6 \][/tex]
