What is the value of $b+c$ if $x^2+bx+c>0$ only when $x\in (-\infty, -2)\cup(3,\infty)$?
That means \(x^2 + bx + c \leqslant 0\) when \(x\in \left[-2, 3\right]\).
And this also means \(x^2 + bx + c < 0\) when \(x\in \left(-2, 3\right)\).
When \(x^2 + bx + c = 0\), \(x = -2\text{ or }x = 3 \)
So \(x^2 + bx + c = (x + 2)(x - 3) = x^2 - x - 6\).
Comparing coefficients, b = -1 and c = -6.
b + c = -1 + (-6) = -7.