[tex]\displaystyle{ E(x)=(3x+1)^{2}-2(x-4)^{2}+(5+x)(5-x)-5x(x+4)+3 }[/tex]
[tex]\displaystyle{ E(x) =9x^{2}+6x+1-2 \cdot(x^{2}-8x+16)+25-x^{2}-5x^{2}-20x + 3 }[/tex]
[tex]\displaystyle{ E(x)=9x^{2}+6x+1-2x^{2}+16x-32+25-x^{2}-5x ^{2}-20x+3 }[/tex]
[tex]\displaystyle{ E(x) = x^{2} \cdot (9 -2-1 - 5) + x \cdot (6 + 16 - 20)+1-32+25+3 }[/tex]
[tex]\displaystyle{ E(x) = x^{2}+2x-3 }[/tex]
[tex]\displaystyle{ E(x) = (x-1)\cdot(x+3) }[/tex]
Q.E.D.
[tex]\displaystyle{ E(2\sqrt{2} -1) = (2\sqrt{2} -1)^{2} + 2 \cdot (2\sqrt{2}-1) - 3 }[/tex]
[tex]\displaystyle{ E(2\sqrt{2}-1) = 4 \cdot 2 - 4\sqrt{2} + 1 + 4\sqrt{2} -2 - 3 }[/tex]
[tex]\displaystyle{ E(2\sqrt{2} -1) = 8 + 1 - 2 - 3 }[/tex]
[tex]\displaystyle{ E(2\sqrt{2}-1) = 4 }[/tex]
4 ∈ Q
Q.E.D.
