Let $g(x)=x^{2}+b x+c$. If $g(2)=0$ and $g(-3)=0$, find $b+c$

Step 1 ：Let \(g(x)=x^{2}+b x+c\). If \(g(2)=0\) and \(g(-3)=0\), find \(b+c\)

Step 2 ：The given function is a quadratic function. The roots of the function are given as 2 and -3.

Step 3 ：We know that the sum of the roots of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(-b/a\) and the product of the roots is given by \(c/a\). Here, \(a=1\), so the sum of the roots is \(-b\) and the product of the roots is \(c\).

Step 4 ：We can set up two equations based on this information and solve for \(b\) and \(c\).

Step 5 ：From the first equation \(-b = -1\), we get \(b = 1\)

Step 6 ：From the second equation \(c = -6\), we get \(c = -6\)

Step 7 ：Finally, we find \(b+c = 1 + (-6) = \boxed{-5}\)