Hello, Dragon!
Some parentheses and spaces would have helped.
I'm guessing what the problem is supposed to be . . .
For what values of 2^x does: .2^x + 2^(x+1) .= .4^x + 4^(x+1)
If I've interpreted the problem correctly, we get an entirely different answer.
We have: .2^x + 2·2^x .= .(2²)^x + (2²)^(x+1}
. . 3·2^x .= .2^(2x) + 2^(2x + 2)
. . 3·2^x .= .2^(2x) + 2²·2^(2x)
. . 3·2^x .= .2^{2x} + 4·2^(2x)
. . 3·2^x .= .5·2^(2x)
Divide by 2^x: .3 .= .5·2^x . **
And we have: .2^x .= .3/5
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** . We can divide by 2^x because 2^x ≠ 0.