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So my main problem:

V is the set of all polynomials of the form at

^{2}+bt+c, where a,b,c are real numbers wtih b=a + 1

(a

_{1}t

^{2}+b

_{1}t+c

_{1}) + (a

_{2}t

^{2}+b

_{2}t+c

_{2}) = (a

_{1}+a

_{2})t

^{2}+(b

_{1}+b

_{2})t+(c

_{1}+c

_{2})

also r*(a

_{1}t

^{2}+b

_{1}t+c

_{1})=ra

_{1}t

^{2}+rb

_{1}t+rc

_{1}

Is it closed?

So I guess I go through every condition and see if it holds off? I know standard polynomials satisify all the conditions. But the book says it is not a vector space becauase b1+b2= (a

_{1}+1)+(a

_{2}+1)= a

_{1}+ a

_{2}+ 2

so our polynomial is not in V and not closed. I don't get what b= a+1 changes. It still seems like every operation/condition is still satisfied. What am I missing? Any help is appreciated.