Given:

Alpha and beta are zeros of polynomial

Polynomial:

[tex]2 {x }^{2} + 4x + 5[/tex]

To find:

[tex] { \alpha }^{2} + { \beta }^{2} [/tex]

Solution

We have

[tex] \alpha + \beta = \frac{ - b}{a} = \frac{ - 4}{2} = - 2[/tex]

also

[tex] \alpha \beta = \frac{c}{a} = \frac{5}{2} = 2.5[/tex]

we also know that

[tex]( { \alpha + \beta )}^{2} = { \alpha }^{2} + { \beta }^{2} + 2 \times \alpha \beta [/tex]

Substituting the given values in the above equation we have:

[tex] {( - 2)}^{2} = { \alpha }^{2} + { \beta }^{2} + 2 \times 2.5[/tex]

Therefore:

[tex] { \alpha }^{2} + { \beta }^{2} = - 1[/tex]