$\text{a)}{7}^{x+1}+7^x=392$  

$7^x\cdot 7+x^x=392$  

$7^x\left(7+1\right)=392$  

$7^x\cdot 8=392\mid :8$  

$7^x=49$  

$7^x=7^2$  

$x=2$  

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$\text{b)}{2}^{x+2}-2^x=96$  

$2^x\cdot 2^2-2^x=96$  

$2^x\left(4-1\right)=96$  

$2^x\cdot 3=96\mid :3$  

$2^x=32$  

$2^x=2^5$  

$x=5$  

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$\text{c)}{3}^{2x-1}+3^{2x-2}-3^{2x-3}=297$  

$3^{2x}\cdot 3^{-1}+3^{2x}\cdot 3^{-2}-3^{2x}\cdot 3^{-3}=297$  

$3^{2x}\left(\frac{1}{3}+\frac{1}{9}-\frac{1}{27}\right)=297$  

$3^{2x}\left(\frac{9}{27}+\frac{3}{27}-\frac{1}{27}\right)=297$  

$3^{2x}\cdot \frac{11}{27}=297\mid \cdot \frac{27}{11}$  

$3^{2x}=729$  

$3^{2x}=3^6$  

$2x=6\mid :2$  

$x=3$  

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$\text{d)}{2}^{x-1}+2^{x-2}+2^{x-3}=448$  

$2^x\cdot 2^{-1}+2^x\cdot 2^{-2}+2^x\cdot 2^{-3}=448$  

$2^x\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\right)=448$  

$2^x\left(\frac{4}{8}+\frac{2}{8}+\frac{1}{8}\right)=448$  

$2^x\cdot \frac{7}{8}=448\mid \cdot \frac{8}{7}$  

$2^x=512$  

$2^x=2^9$  

$x=9$