solve: $18^{x^{2}+4x + 4}=18^{9x + 18}$\n$x=-2$\n$x = 7$\n$x=-2$ and $x = 7$\nno solution

Explanation:

Step1: Use exponential - property

Since the bases are the same ($a^m=a^n$ implies $m = n$ when $a>0,a\neq1$), for $18^{x^{2}+4x + 4}=18^{9x + 18}$, we have $x^{2}+4x + 4=9x + 18$.

Step2: Rearrange to quadratic - form

Rearrange the equation $x^{2}+4x + 4=9x + 18$ to $x^{2}+4x-9x + 4 - 18=0$, which simplifies to $x^{2}-5x - 14=0$.

Step3: Factor the quadratic equation

Factor $x^{2}-5x - 14=0$ as $(x - 7)(x+2)=0$.

Step4: Solve for x

Set each factor equal to zero: $x - 7=0$ gives $x = 7$ and $x+2=0$ gives $x=-2$.

Answer:

$x=-2$ and $x = 7$