<s>[INST] 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 $3^{x^2}\cdot 3^x$?
 [/INST] $3^{x^2+1} = 3^{x^2} \cdot 3$ is correct.
Furthermore,$$ 5^{x^2} \cdot 3 = 3^{x^2} \cdot 5 \iff 5^{x^2-1} = 3^{x^2-1} \iff \Big(\frac{5}{3} \Big)^{x^2-1} = 1$$.
This happens exactly when $x^2-1=0$,or when $x = \pm 1$. 
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