PROBLEM 1. a) Factorize $3 x^{2}+4 \sqrt{6} x+8-y^{2}$.

b) Find the minimum and maximum of the expression $\frac{x+1}{|x+1|}+\frac{|x+2|}{x+2}+\frac{x+3}{|x+3|}$, for $x \in \mathbb{R} \backslash\{-3,-2,-1\}$.
## PROBLEM 1.

$(2 p)$
a) $E=3 x^{2}+4 \sqrt{6} x+8-y^{2}=(x \sqrt{3}+2 \sqrt{2})^{2}-y^{2}$;

$(2 p)$ Completion, $E=(x \sqrt{3}+2 \sqrt{2}-y)(x \sqrt{3}+2 \sqrt{2}+y)$.

$(1 p)$

b) The fractions can only take the values 1 or -1

(1p) The minimum value of the expression is -3

(1p) The maximum value of the expression is 3