You should be aware of the fundamental theorem of algebra (FTCoA). If a quadratic has roots at
and
, then we can factorize it as

Expanding, this is equivalent to

Now, if we can also write this

then we must have

(By the way, these are known as Vieta's formulas.)
We're told that the quadratic,

has roots
and
, so by the FTCoA, we can write

Expanding this last form, we have the identity and exact values

(i) We want to construct a new quadratic
such that its roots are
and
. By the FTCoA, we would have

Expanding,

Now, notice that


It follows that

which we can rewrite with integer coefficients by scaling each term by a factor of 9 to get

(ii) Now we want a quadratic
with roots at
and
. This means


or more cleanly, scaling by 3,

(iii) Now the roots are
and
, which gives



(iv) If the roots are
and
, we have

Note that



