<s>[INST] Extending Completing the square to expressions of $ (x+y+z)^2$ We all know expressions such as $x^2+14x+49 = (x+7)^2$ because it is easily recognizable as a perfect square. What about a expressions in the form of $x^2+2xy+2yz+2xz+y^2+z^2=(x+y+z)^2$
The question is how do I factor $0.09e^{-2t}+0.24e^{-t}+0.34+0.24e^t+0.09e^{2t}$ into the form of $(x+y+z)^2$?
 [/INST] Setting $e^t=a,$ we have $\displaystyle\frac{9+24a+34a^2+24a^3+9a^4}{100a^4}$ which can be written as $\displaystyle\frac{(3a^2+ab+3)^2}{(100a^2)^2}$ where $b$ is independent of $a$
$\displaystyle\implies 9+24a+34a^2+24a^3+9a^4=9a^4+a^2b^2+9+6ba^3+6ba+18a^2$
$\displaystyle\implies 9+24a+34a^2+24a^3+9a^4=9a^4+6ba^3+a^2(18+b^2)+6ba+9$
Comparing the coefficients of $a$ or $a^3,\displaystyle24=6b\iff b=4$
Comparing the coefficients of $a^2,\displaystyle34=18+b^2\iff b^2=16\iff b=\pm4$
So, $b=?$
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