Let \( p(x)=x^{2}+bx+c \) be a quadratic polynomial with real coefficients such that \( \int\limits_{0}^{1} p(x)\,dx = 1 \) and \( p(x) \) leaves remainder \( 5 \) when it is divided by \( (x-2) \). Then the value of \( 9(b+c) \) is equal to:
\( 7 \)
\( 15 \)
\( 9 \)
\( 11 \)
