In the equation $4^{x+2}=2^{x+3}+48$, the value of  $x$ will be

$4^{x+2}=2^{x+3}+48$

$2^{2(x+2)}=2^x\times 2^3+48$

$2^{2x+4}=2^x\times 8+48$

${\left(2^x\right)}^2\times 2^4=2^x\times 8+48$

${\left(2^x\right)}^2\times 16=8(2^x+6)$

$2.{\left(2^x\right)}^2-2^x-6=0$

$2a^2-a-6=0$

$2a^2+3a-4a-6=0$

$a(2a+3)-2(2a+3)=0$

$(2a+3)(a-2)=0$

$a=2\text{\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}}-\frac{3}{2}$

$2^x=2\Rightarrow x=1$

$2^x=-\frac{3}{2}$ does not exists.