$x^3+3x^2+4x+5=0$ and $x^3+2x^2+7x+3=0$, how many common roots they have?

Question

My attempt, Equate both, at the end you will get $x^2-3x-2=0$

That means $x=-1$ and $x=2$. But what after that. Please provide solutions as well.

Answer 1

Compute the $\gcd$ of the two polynomials to see that is $1$. Thus, no common (non-constant) factor, no common roots.

Answer 2

You have a mistake in your calculation (or a typo). $-1,2$ are not the roots of $x^{2}-3x-2$.

Denote $$ p(x)=x^{3}+3x^{2}+4x+5$$ $$ q(x)=x^{3}+2x^{2}+7x+3 $$

and $x_{0}$ is such that $$ p(x_{0})=q(x_{0})=0 $$

then $$ p(x_{0})-q(x_{0})=0 $$

as well. This gives us $$ x_{0}^{2}-3x_{0}+2=0 $$

and since the zeros of the polynomial $$ x^{2}-3x+2 $$

are $1$ and $2$ those are the only possible values for $x_{0}$.

Note: $$ p(x_{0})-q(x_{0})=0\iff x_{0}^{2}-3x_{0}+2=0\iff x\in\{1,2\} $$

But none are roots of $p,q$ and so those don't have any roots in common