<s>[INST] Pattern in factorising monic Quadratics I have noticed a pattern while factoring quadratic expressions in the form: $x^2+(a+1)x+a$ ($a$ is a constant). For example, 


*

*$x^2+3x+2=(x+2)(x+1)$ 

*$x^2+92x+91=(x+91)(x+1)$

*$x^2+5x+4=(x+4)(x+1)$


The pattern which can be assumed here is:
$$x^2+(a+1)x+a=(x+a)(x+1)$$
My question is: How do you prove this identity?
 [/INST] A couple of ways other than direct factorization:


*

*by the quadratic formula the roots of $\,x^2+(a+1)x+a\,$ are $$\,x_{1,2}=\dfrac{-(a+1)\pm\sqrt{(a+1)^2-4a}}{2}=\dfrac{-(a+1)\pm\sqrt{(a-1)^2}}{2}=\begin{cases}-1 \\-a\end{cases}\,$$and therefore the quadratic factors as $\,(x-x_1)(x-x_2)=(x+1)(x+a)\,$;

*two quadratics that have equal values at three different points are identical (since their difference would be a polynomial of degree $\le 2$ with $3$ roots), so it is enough to verify that $\,x^2+(a+1)x+a=(x+1)(x+a)\,$ for three values of $x$, for example $x=0,-1,-a$.
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