Hello, an exercise require to simplify this equation but i don't know what they are, could you help me? thank you!
$\displaystyle \left\frac{a^2+4a\sqrt3+12}{2a^2+3a\sqrt3-6}$
$\displaystyle (a+ b)(a+ c)= a^2+ (b+c)x+ bc$
so to factor $\displaystyle x^2+ 4a\sqrt{3}+ 12$
you are looking for two numbers that add to $\displaystyle 4\sqrt{3}$ and multiply to 12. $\displaystyle 2\sqr{3}$ and $\displaystyle 2\sqrt{3}$ work!
To factor $\displaystyle 2a^2+ 3a\sqrt{3}- 6$, try $\displaystyle (a- 2\sqrt{3})(ba+ c)= ba^2+(b- 2\sqrt{3}c)a- 2c\sqrt{3}= 2a^2+ 3a\sqrt{3}- 6$ and compare coefficients.