We can simplify the expression $(\sqrt{5} - 3)(\sqrt{5} + 2)$ by applying the distributive property (also known as the FOIL method for binomials).

The general rule for multiplying binomials $(a + b)(c + d)$ is: $ac + ad + bc + bd$ Here, the terms are:

• $a = \sqrt{5}$,
• $b = -3$,
• $c = \sqrt{5}$,
• $d = 2$.

Now, let's apply this:

$(\sqrt{5} - 3)(\sqrt{5} + 2) = \sqrt{5} \cdot \sqrt{5} + \sqrt{5} \cdot 2 - 3 \cdot \sqrt{5} - 3 \cdot 2$

Now, simplify each term:

1. $\sqrt{5} \cdot \sqrt{5} = 5$,
2. $\sqrt{5} \cdot 2 = 2\sqrt{5}$,
3. $-3 \cdot \sqrt{5} = -3\sqrt{5}$,
4. $-3 \cdot 2 = -6$.

Putting it all together: $5 + 2\sqrt{5} - 3\sqrt{5} - 6$

Now, combine the like terms: $5 - 6 + (2\sqrt{5} - 3\sqrt{5}) = -1 - \sqrt{5}$

Thus, the simplified expression is: $-1 - \sqrt{5}$

Would you like further clarification or more examples?

Here are some related questions:

1. How do you simplify $(\sqrt{2} - 4)(\sqrt{2} + 3)$?
2. What happens if we multiply $(a - b)(a + b)$?
3. How can you simplify expressions involving square roots in a more general case?
4. How do you perform binomial multiplication with irrational numbers?
5. What is the expanded form of $(x + 5)(x - 3)$?

Tip: When multiplying binomials, always remember to combine like terms and keep track of the signs carefully!