If α and β are the zeroes of quadratic polynomial x²-2x+3=0. Find . . .

Question : If α and β are the zeroes of quadratic polynomial x²-2x+3=0. Find a polynomial whose zeros are α+2 and β+2 .

Doubt by Yathartha

Solution : 

p(x)=x²-2x+3
a=1

b=-2

c=3

Sum of Zeroes

α+β = -b/a

α+β = -(-2)/1

α+β = 2

Product of Zeroes

αβ = c/a

αβ = 3/1

αβ = 3

For New Polynomial 

Sum of Zeroes (S) 
S= (α+2)+(β+2)
S= α+β+4
S= 2+4
S= 6

Product of Zeroes (P)
P= (α+2)(β+2)
P= αβ+2α+2β+4
P= αβ+2(α+β)+4
P= 3+2(2)+4
P= 3+4+4
P=11

New Polynomial will be given by 

p'(x)=x²-Sx+P
p'(x)=x²-(6)x+11

p'(x)=x²-6x+11

which is the required polynomial.