Answer:
[tex]\boxed{\sf \ \ \ 81x^2-270x-529\ \ \ }[/tex]
Step-by-step explanation:
hello
the quadratic equation whose roots are
[tex]a^2+1 \ \ \ and \ \ \ b^2+1[/tex]
is like
[tex](x-a^2-1)(x-b^2-1)=x^2-(a^2+b^2+2)x-(a^2+1)(b^2+1)[/tex]
as a and b are the roots of
[tex]3x^2-6x-2=3(x^2-2x-\dfrac{2}{3})[/tex]
a+b = 2
ab=-2/3
so
[tex]a^2+b^2+2=(a+b)^2-2ab-2=2^2+\dfrac{2*2}{3}-2=2+\dfrac{4}{3}=\dfrac{10}{3}[/tex]
[tex](a^2+1)(b^2+1)=(ab)^4+(a+b)^2-2ab+1=\dfrac{2^4}{3^4}+2^2+\dfrac{4}{3}+1[/tex]
[tex]\dfrac{16+5*81+4*27}{81}=\dfrac{529}{81}[/tex]
so the polynomial is like
[tex]x^2-\dfrac{10}{3}x-\dfrac{529}{81}[/tex]
let's multiply by 81 to have integer coefficients
[tex]81x^2-270x-529[/tex]
hope this helps
