Question

URGENT !!! 25 POINTS AND BRAINIEST FOR CORRECT ANSWER!!!

Answer 1

Given a polynomial p(x) with leading coefficient $a$ and solutions $x_1,x_2,\ldots,x_n$, we have

$p(x)=a(x-x_1)(x-x_2)\ldots(x-x_n)$

That's why we have to find the solutions first. We complete the square by adding 3 to both sides:

$x^2-2x-2=0 \iff (x^2-2x-2)+3=3 \iff x^2-2x+1=3 \iff (x-1)^2=3$

From here, we continue by taking the square root of both sides:

$x-1=\pm 3 \iff x=1\pm\sqrt{3}$

So, the two solutions are

$x_1=1+\sqrt{3},\quad x_2=1-\sqrt{3}$

Which yields the factorization

$x^2-2x-2=(x-1-\sqrt{3})(x-1+\sqrt{3})$