<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] We know that $$4x^4-16x^3-17x^2+27x-7 =(4x^2-20x+7)(x^2+x-1) $$ $$=(x- \frac{5+3\sqrt {2}}{2})(x-\frac {5-3\sqrt {2}}{2})(x-\frac {-1 +\sqrt {5}}{2})(x+\frac {-1-\sqrt {5}}{2}) $$
Observe that $\frac {5+3\sqrt {2}}{2} $ is a root of this polynomial. Also observe that the above polynomial is $4$ less than the required polynomial in question. Thus, the answer is $4$. Hope it helps. 
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