[tex]\displaystyle\\
Ex. ~6)\\\\
b)\\
A_{x+2}^2 =56(x+2) \\
(x+2)(x+1) = 56(x+2) ~~~| :(x+2)\\
x+1 = 56\\
x = 56 - 1 = \boxed{55}\\\\
c)\\
A_{x+1}^2 =30 \\
(x+1)\cdot x=30\\
x(x+1)=30\\
\text{x si (x+1) sunt numere naturale consecutive.}\\
\text{Descompunem pe 30 in produs de numere consecutive.}\\
x(x+1)=5\cdot 6\\
\Longrightarrow~~~ x = \boxed{5}
[/tex]
[tex]\displaystyle\\
d)\\
A_{n-1}^5 =18\cdot A_{n-3}^4 \\
(n-1)(n-2)(n-3)(n-4)(n-5) = 18(n-3)(n-4)(n-5)(n-6)\\
\text{Impartim ecuatia la: }(n-3)(n-4)(n-5) \\
(n-1)(n-2) = 18(n-6)\\
n^2 -3n +2 = 18n - 108\\
n^2 -3n-18n + 2+108 = 0\\
n^2 -21n +110 =0\\
n^2 -10n - 11n +110 = 0\\
n(n-10) -11(n-10)= 0\\
(n-10)(n-11)=0\\
n_1 = \boxed{10}\\
n_2 = \boxed{11}[/tex]
[tex]\displaystyle\\
e)\\
A_x^2+2A_{x+1}^2=30\\
x(x-1) + 2(x+1)\cdot x = 30\\
x(x-1) + 2x(x+1) = 30\\
x^2-x + 2x^2 + 2x=30\\
3x^2 +x = 30\\
3x^2 +x - 30=0\\
3x^2 -9x + 9x +x-30 = 0\\
3x^2 -9x + 10x -30 = 0\\
3x(x-3)+ 10(x-3)=0\\
(x-3)(3x+10)=0\\
x-3 = 0 ~~~\Longrightarrow ~~~x = \boxed{3}\\
3x+10=0~~~\Longrightarrow ~~~x \notin N[/tex]
[tex]\displaystyle\\
f)\\
A_{x+1}^{x-1}=360\\\\
\frac{P_{x+1}}{P_2}=360\\\\
\frac{(x+1)!}{2}=360\\\\
(x+1)! = 360 \cdot 2\\
(x+1)! = 720\\
(x+1)! = 6!\\
x+1 = 6\\
x = 6-1 = \boxed{5}[/tex]
[tex]\displaystyle\\
g)\\
A_{x+3}^{x+2}=10A_{x+2}^{x+1}\\\\
\frac{P_{x+3}}{P_1} = 10\cdot \frac{P_{x+2}}{P_1}\\\\
\frac{(x+3)!}{1} = 10\cdot \frac{(x+2)!}{1} \\\\
(x+3)! = 10\cdot (x+2)!\\\\
\frac{(x+3)!}{(x+2)!} =10\\\\
\frac{(x+2)! \cdot (x+3)}{(x+2)!} =10\\\\
x+3 = 10
x = 10 - 3 = \boxed{7}[/tex]
