Question

1. $-3sqrt{15}(4 + sqrt{6})$ 12) $sqrt{3}(-sqrt{3} + sqrt{2})$ \\(4sqrt{2} - 4)(sqrt{2} + 1) 14) $(-4sqrt{2} + 5)(sqrt{2} - 5)$ \\$(sqrt{5} + sqrt{2})(-4sqrt{5} + 4sqrt{4})$ 16) $(2 + sqrt{2})(-3 + sqrt{2})$

Explanation:

Step1: Distribute $-3\sqrt{15}$

$-3\sqrt{15} \times 4 + (-3\sqrt{15}) \times \sqrt{6}$
$=-12\sqrt{15} - 3\sqrt{15 \times 6}$
$=-12\sqrt{15} - 3\sqrt{90}$
$=-12\sqrt{15} - 3 \times 3\sqrt{10}$
$=-12\sqrt{15} - 9\sqrt{10}$

Step2: Distribute $\sqrt{3}$

$\sqrt{3} \times (-\sqrt{3}) + \sqrt{3} \times \sqrt{2}$
$=-(\sqrt{3})^2 + \sqrt{3 \times 2}$
$=-3 + \sqrt{6}$

Step3: Use FOIL method

$4\sqrt{2} \times \sqrt{2} + 4\sqrt{2} \times 1 - 4 \times \sqrt{2} - 4 \times 1$
$=4 \times 2 + 4\sqrt{2} - 4\sqrt{2} - 4$
$=8 - 4 + (4\sqrt{2} - 4\sqrt{2})$
$=4 + 0 = 4$

Step4: Use FOIL method

$-4\sqrt{2} \times \sqrt{2} + (-4\sqrt{2}) \times (-5) + 5 \times \sqrt{2} + 5 \times (-5)$
$=-4 \times 2 + 20\sqrt{2} + 5\sqrt{2} - 25$
$=-8 - 25 + (20\sqrt{2} + 5\sqrt{2})$
$=-33 + 25\sqrt{2}$

Step5: Simplify $\sqrt{4}$ first, then FOIL

$\sqrt{4}=2$, so the expression becomes $(\sqrt{5}+\sqrt{2})(-4\sqrt{5}+4 \times 2)=(\sqrt{5}+\sqrt{2})(-4\sqrt{5}+8)$
$\sqrt{5} \times (-4\sqrt{5}) + \sqrt{5} \times 8 + \sqrt{2} \times (-4\sqrt{5}) + \sqrt{2} \times 8$
$=-4 \times 5 + 8\sqrt{5} - 4\sqrt{10} + 8\sqrt{2}$
$=-20 + 8\sqrt{5} - 4\sqrt{10} + 8\sqrt{2}$

Step6: Use FOIL method

$2 \times (-3) + 2 \times \sqrt{2} + \sqrt{2} \times (-3) + \sqrt{2} \times \sqrt{2}$
$=-6 + 2\sqrt{2} - 3\sqrt{2} + 2$
$=(-6 + 2) + (2\sqrt{2} - 3\sqrt{2})$
$=-4 - \sqrt{2}$

Answer:

1. $\boldsymbol{-12\sqrt{15} - 9\sqrt{10}}$
2. $\boldsymbol{-3 + \sqrt{6}}$
3. $\boldsymbol{4}$
4. $\boldsymbol{-33 + 25\sqrt{2}}$
5. $\boldsymbol{-20 + 8\sqrt{5} + 8\sqrt{2} - 4\sqrt{10}}$
6. $\boldsymbol{-4 - \sqrt{2}}$