Let \( P(x)=x^{2}+b x+c \) be a quadratic polynomial with real coefficients such that \( \int_{0}^{1} P(x) d x=1 \) and …

Let \( P(x)=x^{2}+b x+c \) be a quadratic polynomial with real coefficients such that \( \int_{0}^{1} P(x) d x=1 \) and \( P(x) \) leaves remainder 5 when it is divided by \( (x-2) \). Then the value of \( 9(b+c) \) is equal to:

[JEE Main 2021, 16 Mar (Shift 2)]