solve: $12^{x^{2}+5x - 4}=12^{2x + 6}$\n$x = 2$\n$x=-5$\n$x = 2,x=-5$\nno solution

Explanation:

Step1: Set exponents equal

Since the bases are the same ($a^m=a^n$ implies $m = n$ for $a>0,a\neq1$), we set $x^{2}+5x - 4=2x + 6$. $x^{2}+5x-4=2x + 6$

Step2: Rearrange to quadratic form

Move all terms to one - side to get a quadratic equation. $x^{2}+5x-2x-4 - 6=0$ $x^{2}+3x - 10=0$

Step3: Factor the quadratic

Factor the quadratic equation $x^{2}+3x - 10$. $(x + 5)(x - 2)=0$

Step4: Solve for x

Set each factor equal to zero. If $x+5=0$, then $x=-5$; if $x - 2=0$, then $x = 2$.

Answer:

$x = 2,x=-5$