<s>[INST] Compute the following without the calculator 
$$4\left(5+3\sqrt2\over 2\right)^4-16\left(5+3\sqrt2\over 2\right)^3-17\left(5+3\sqrt2\over 2\right)^2+27\left(5+3\sqrt2\over 2\right)-3$$

Please solve the following equation without using calculator.
Substituting $\left(5+3\sqrt2\over 2\right)$ to x must be the first step, but then I don't know how to factor it.
 [/INST] Let $f(x)=4x^4-16x^3-17x^2+27x-3$. Then $g(x)=f(x)-4=(4x^2 - 20x + 7)(x^2 + x - 1)$.
Since $a=(5+3\sqrt{2})/2$ satisfies $4a^2-20a+7=0$ we know that $g(a)=0$ and hence
$f(a)=4$. So the answer is $4$.
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