Giải thích các bước giải:

1) $x^{2} - 1 = \left ( x - 1 \right )\left ( x + 1 \right )$

2) $x^{2} - 2 = x^{2} - \left ( \sqrt{2} \right )^{2} = \left ( x - \sqrt{2} \right )\left ( x + \sqrt{2} \right )$

3) $x^{2} - 3 = x^{2} - \left ( \sqrt{3} \right )^{2} = \left ( x - \sqrt{3} \right )\left ( x + \sqrt{3} \right )$

4) $5 - 2x^{2} = 2\left ( \dfrac{5}{2} - x^{2} \right ) = 2\left [ \left ( \sqrt{\dfrac{5}{2}} - x^{2} \right ) \right ] = 2\left ( \sqrt{\dfrac{5}{2}} - x \right )\left ( \sqrt{\dfrac{5}{2}} + x \right )$

5) $x^{2} + 2\sqrt{2}x + 2 = x^{2} + 2\sqrt{2}x + \left ( \sqrt{2} \right )^{2} = \left ( x + \sqrt{2} \right )^{2}$

6) $x^{2} - 2\sqrt{3}x + 3 = x^{2} - 2\sqrt{3}x + \left ( \sqrt{3} \right )^{2} = \left ( x - \sqrt{3} \right )^{2}$

7) $x - 4\sqrt{x} + 4 = \left ( \sqrt{x} \right )^{2} - 2.2\sqrt{x} + 2^{2} = \left ( \sqrt{x} - 2 \right )^{2}$

8) $x + 2\sqrt{x} + 1 = \left ( \sqrt{x} \right )^{2} - 2.1\sqrt{x} + 1^{2} = \left ( \sqrt{x} - 1 \right )^{2}$

9) $x - 1 = \left ( \sqrt{x} \right )^{2} - 1^{2} = \left ( \sqrt{x} - 1 \right )\left ( \sqrt{x} + 1 \right )$

10) $9x - 4 = 9\left ( x - \dfrac{4}{9} \right ) = 9\left [ \left ( \sqrt{x} \right )^{2} - \left ( \dfrac{2}{3} \right )^{2} \right ] = 9\left ( \sqrt{x} - \dfrac{2}{3} \right )\left ( \sqrt{x} + \dfrac{2}{3} \right )$

11) $a + 3 = 3 - a$ (vì $a < 0$)

$= \left ( \sqrt{3} \right )^{2} - \left ( \sqrt{a} \right )^{2}$

$= \left ( \sqrt{3} - \sqrt{a} \right )\left ( \sqrt{3} + \sqrt{a} \right )$