Explanation:

Since, α and β are the roots of the quadratic equation, x2+x−1=0.
∴α+β=−1&α⋅β=−1.
We have to find an equation whose roots are α5 and β5.
Now, product of the roots of the required equation is,
=α5⋅β5
=(αβ)5
=(−1)5
=−1.
And, sum of the roots of the required equation is,
=α⁵+β⁵
=(α³+β³)(α²+β²)−α³β²−β³α²
={(α+β)³−3αβ(α+β)}⋅{(α+β)²−2αβ}−α²β²(α+β)
={(−1)³−3(−1)(−1)}⋅{(−1)²−2(−1)}−(−1)²(−1)
=(−4)×(3)+1
=−11
So, the required equation is,
x²−(sum of the roots)x+(product of the roots)=0,
i.e., x²+11x−1=0.