2. Suppose that the equation $$x ^ { 4 } + A x ^ { 2 } + B = \left( x ^ { 2 } + a x + b \right) \left( x ^ { 2 } - a x + b \right)$$ holds for all values of $x$. (i) Find $A$ and $B$ in terms of $a$ and $b$. (ii) Use this information to find a factorization of the expression $$x ^ { 4 } - 20 x ^ { 2 } + 16$$ as a product of two quadratics in $x$. (iii) Show that the four solutions of the equation $$x ^ { 4 } - 20 x ^ { 2 } + 16 = 0$$ can be written as $\pm \sqrt { 7 } \pm \sqrt { 3 }$.
2. Suppose that the equation
$$x ^ { 4 } + A x ^ { 2 } + B = \left( x ^ { 2 } + a x + b \right) \left( x ^ { 2 } - a x + b \right)$$
holds for all values of $x$.\\
(i) Find $A$ and $B$ in terms of $a$ and $b$.\\
(ii) Use this information to find a factorization of the expression
$$x ^ { 4 } - 20 x ^ { 2 } + 16$$
as a product of two quadratics in $x$.\\
(iii) Show that the four solutions of the equation
$$x ^ { 4 } - 20 x ^ { 2 } + 16 = 0$$
can be written as $\pm \sqrt { 7 } \pm \sqrt { 3 }$.\\