\(2^x+2+2^x+1+2^x=224\)
\(\Leftrightarrow3\left(x^2\right)+3=224\)
\(\Leftrightarrow3\left(x^2+1\right)=224\)
\(\Leftrightarrow x^2+1=\frac{224}{3}\)
\(\Leftrightarrow x^2=\frac{221}{3}\)
\(\Leftrightarrow x=\sqrt{\frac{221}{3}}=\frac{\sqrt{663}}{3}\)