What is the least possible degree of polynomial with real coefficient having 2ω², 3 + 4ω, 3 + 4ω², 5 - ω - ω² as roots

First of all, we should not think that because of the 4 roots, the degree of the polynomial will be 4.

We need to remember that complex roots always occur in pairs.

Now,

Now, to find the value of the given pair roots,

1) 2ω² = -1 - √3 i

∴ -1 + √3i will also be a root.

2) 3 + 4ω = 3 - 2 + 2√3i

= 1 + 2√3i

∴ 1 - 2√3i will also be a root.

3) 3 + 4ω² = 3 - 2 - 2√3i

= 1 - 2√3i

i.e, 3 + 4ω and 3 + 4ω² are conjugate pairs.

= 6

i.e. 6 is a root.

Total we have 5 roots.

The minimum degree of the polynomial is 5.