If t = 2^x + 1, then in terms of t, 4^x is

Let's first write \(4^x\) as a power of \(2\) (like we have in the other equation)
Replace \(4\) with \(2^2\) to get: \(4^x = (2^2)^x= 2^{2x}\)
So we now want to find a value that is equivalent to \(2^{2x}\)

Now let's do something with the first equation.
Take: \(t = 2^{x + 1}\)
Rewrite as: \(t = (2^x)(2^1)\)
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Aside: Some students won't like that last step, so here's why it works:
If we take \((2^x)(2^1)\) and find the product by using the product law we get: \(2^{x+1}\)
So as you can see that last step was mathematically legitimate
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Simplify to get: \(t = (2^x)(2)\)
Divide both sides by \(2\) to get: \(\frac{t}{2} = 2^x\)

We're almost there!
We want to find a value that is equivalent to \(2^{2x}\)

If we take: \(\frac{t}{2} = 2^x\)
And square both sides we get: \((\frac{t}{2})^2 = (2^x)^2\)
Simplify to get: \(\frac{t^2}{4}= 2^{2x}\)

Answer: E

Cheers,
Brent