To find a polynomial whose zeroes are 2α + 1 and 2β + 1, given that α and β are the zeroes of the quadratic polynomial p(x) = x^2 + x - 2, we can use the property of zeroes of a polynomial.
If α and β are the zeroes of a polynomial p(x), then the polynomial whose zeroes are (aα + b) and (aβ + b) is given by:
p'(x) = a^2 p((x-b)/a)
In this case, we have a = 1, b = 1, and p(x) = x^2 + x - 2.
Substituting these values in the above formula, we get:
p'(x) = (1)^2 p((x-1)/1) p'(x) = (x-1)^2 + (x-1) - 2 p'(x) = x^2 - 2x - 1
Therefore, the polynomial whose zeroes are 2α + 1 and 2β + 1 is p'(x) = x^2 - 2x - 1.
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