Simplify (2+ square root of 5)(-2+ square root of 5k)

$(2 + \sqrt{5}) (- 2 + \sqrt{5 k})$

Expand $(2 + \sqrt{5}) (- 2 + \sqrt{5 k})$ using the FOIL Method.

Apply the distributive property.

$2 (- 2 + \sqrt{5 k}) + \sqrt{5} (- 2 + \sqrt{5 k})$

Apply the distributive property.

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

Apply the distributive property.

$2 \cdot - 2 + 2 \sqrt{5 k} + \sqrt{5} \cdot - 2 + \sqrt{5} \sqrt{5 k}$

Simplify each term.

Multiply $2$ by $- 2$ .

$- 4 + 2 \sqrt{5 k} + \sqrt{5} \cdot - 2 + \sqrt{5} \sqrt{5 k}$

Move $- 2$ to the left of $\sqrt{5}$ .

$- 4 + 2 \sqrt{5 k} - 2 \cdot \sqrt{5} + \sqrt{5} \sqrt{5 k}$

Multiply $\sqrt{5} \sqrt{5 k}$ .

Combine using the product rule for radicals.

$- 4 + 2 \sqrt{5 k} - 2 \sqrt{5} + \sqrt{5 (5 k)}$

Multiply $5$ by $5$ .

$- 4 + 2 \sqrt{5 k} - 2 \sqrt{5} + \sqrt{25 k}$

Rewrite $25$ as $5^{2}$ .

$- 4 + 2 \sqrt{5 k} - 2 \sqrt{5} + \sqrt{5^{2} k}$

Pull terms out from under the radical.

$- 4 + 2 \sqrt{5 k} - 2 \sqrt{5} + 5 \sqrt{k}$