Solve for $x$ in $3\cdot 5^{x^2}+3^{x^2+1}-8\cdot 3^{x^2}=0$ $3\cdot 5^{x^2}+3^{x^2+1}-8\cdot 3^{x^2}=0$
Note that I didn't forget any parenthesis. I copied it as it is in my book.
I tried:
$$3\cdot 5^{x^2}+3^{x^2+1}-8\cdot 3^{x^2}=0 \Leftrightarrow 3\cdot 5^{x^2} - 5 \cdot 3^{2x} = 0 \Leftrightarrow 5 \cdot 3^{x^2} = 3 \cdot 5^{x^2} \Leftrightarrow ???$$
What do I do next?
Also, this $3^{x^2+1}$ confuses me. Does it expand to $3^{x^2}\cdot 3$ or $3^{x^2}\cdot 3^x$?

$3^{x^2+1}$ indeed equals $3\cdot 3^{x^2}$. Proceeding from your last equality:
$$
5\cdot 3^{x^2}=3\cdot 5^{x^2}\iff 3^{x^2-1}=5^{x^2-1}\iff(x^2-1)\log(3/5)=0.
$$
The last $\iff$ above is obtained by taking $\log$ and simplifying. Can you continue from here?
