Solving $2^{2x+1} - 2^{x+4} = 2^3 - 2^x$ $$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$
How can I solve an exponential equation that has many terms as the one above. Include more than one method if available.

The key here is to transform this into a quadratic equation.
$$2^{2x+1} - 2^{x+4} = 2^3 - 2^x$$ can be rewritten as $$2 \cdot 2^{2x} - 2^4 \cdot 2^x = 2^3 -2^x.$$
Temporarily we will express $2^x=u$ and we find that the equation turns into $$2u^2 - 16u = 8 - u.$$ Finally writing this as $$2u^2 - 15u - 8 = 0$$ we can factor or use the quadratic equation to solve for $u$. Then remembering that $u=2^x$ we can find the solution for $x$ by writing $\log_2(u)=\ln(u)/\ln(2)=x$. Just be sure to try plugging the answers back into the original equation at the end.
