Work Step by Step
We have $$ p(x)=x^2+ax+b, \quad p'(x)= 2x+a.$$ Since, $ p(1)=0$ and $ p'(1)=4$, we get the equations $$ a+b=-1, \quad a=2$$ Hence we have the solution $ a=2, b=-3$. Then, $ p(x)=x^2+2x-3$.
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