<s>[INST] Simplify the expression: $(2x+1)×\left(x^2+(x+1)^2 \right)×\left( (x^4+(x+1)^4) \right)×...×\left(x^{64}+(x+1)^{64}\right)$ Simplify the expression:
$$(2x+1)×\left(x^2+(x+1)^2 \right)×\left( (x^4+(x+1)^4) \right)×...×\left(x^{64}+(x+1)^{64}\right)$$
I used the general method:
$(2x+1)(2x^2+2x+1)(2x^4+4x^3+6x^2+4x+1)×...$
But, But I'm stuck here.
 [/INST] We can rewrite the product as a telescoping one. The end result is:
$$\prod_{k=0}^6 \left((1+x)^{2^k} + x^{2^k}\right) =
\prod_{k=0}^6 \frac{(1+x)^{2^{k+1}} - x^{2^{k+1}}}{(1+x)^{2^k} - x^{2^k}}
= \frac{(1+x)^{2^7} - x^{2^7}}{(1+x)^{2^0} - x^{2^0}} = (x+1)^{128} - x^{128}$$
</s>