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By the polynomial remainder theorem, the first polynomial remainder is $f(1)=23-2p$. Equating it to $9$ gives $p=7$.

For the separate polynomial long division, successively cancel the leading powers of the dividend. Subtracting $x^2(x-1)$ leaves $2x^2+2x+1$; subtracting $2x(x-1)$ leaves $4x+1$; subtracting $4(x-1)$ leaves $5$. ThusThe polynomial remainder also equals the dividend evaluated at $x=1$.

The derivative of the parabola is $dy/dx=2+2x$, giving tangent line slope $4$ at $(1,3)$. A normal line to a plane curve has the negative reciprocal slope, since its direction is orthogonal to a tangent vector. HenceThe point $(1,3)$ satisfies the equation, and the product of the two slopes is $-1$.