\(5^{x+1}+5^{x+2}=750\)
\(\Rightarrow5^{x+1}\left(1+5\right)=750\)
\(\Rightarrow5^{x+1}.6=750\Rightarrow5^{x+1}=125\)
\(\Rightarrow5^{x+1}=5^3\)
\(\Rightarrow x+1=3\Rightarrow x=2\)
\(5^{x+1}+5^{x+2}=750\)
\(\Rightarrow5^x.5+5^x.5^2=750\)
\(\Rightarrow5^x.5+5^x.25=750\)
\(\Rightarrow5^x\left(5+25\right)=750\)
\(\Rightarrow5^x.30=750\)
\(\Rightarrow5^x=25\)
\(\Rightarrow x=2\)