7) Certainly there is a typo in the statement, just see that the expression of item (ii) is different from that of item (i). Probably the correct expression is: \(2x^2-4x+5\). With this consideration, we can continue.
(i) Let E the expression that we are analyzing:
\(
E=2x^2-4x+5\\\\
E=2x^2-4x+2-2+5\\\\
E=2(x^2-2x+1)-2+5\\\\
E=2(x-1)^2+3
\)
Since (x-1)² is a perfect square, it is a positive number. So, E is a result of a sum of two positive numbers, 2(x-1)² and 3. Hence, E is a positive number, too.
(ii) Manipulating the expression:
\(2x^2+5=4x\\\\
2x^2-4x+5=0\)
So, it's the case when E=0. However, E is always a positive number. Then, there is no real number x that satisfies the expression.
8) Let E the expression that we want to calculate:
\(E=(2+1)(2^2+1)(2^4+1)\cdot ...\cdot(2^{32}+1)+1\\\\
E-1=(2+1)(2^2+1)(2^4+1)\cdot ...\cdot(2^{32}+1)\)
Multiplying by (2-1) in the both sides:
\((2-1)(E-1)=(2-1)(2+1)(2^2+1)(2^4+1)\cdot ...\cdot(2^{32}+1)\\\\
(E-1)=\underbrace{(2-1)(2+1)}_{2^2-1}(2^2+1)(2^4+1)\cdot ...\cdot(2^{32}+1)\\\\
(E-1)=\underbrace{(2^2-1)(2^2+1)}_{2^4-1}(2^4+1)\cdot ...\cdot(2^{32}+1)\\\\ ...\)
Repeating the process, we obtain:
\(...\\\\ E-1=(2^{32}-1)(2^{32}+1)\\\\
E-1=2^{64}-1\\\\
\boxed{E=2^{64}}\)
