1. The problem is to understand the quadratic expression $x^2 + bx + c$ and how to work with it. 2. This is a quadratic polynomial where $b$ and $c$ are constants. 3. The general form of a quadratic equation is $ax^2 + bx + c = 0$, here $a=1$. 4. Important rules: The solutions (roots) of the quadratic can be found using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 5. Since $a=1$, the formula simplifies to: $$x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$$ 6. The discriminant $\Delta = b^2 - 4c$ tells us the nature of the roots: - If $\Delta > 0$, two distinct real roots. - If $\Delta = 0$, one real root (repeated). - If $\Delta < 0$, two complex roots. 7. To factor the quadratic (if possible), find two numbers that multiply to $c$ and add to $b$. 8. Example: If $b=5$ and $c=6$, factors of 6 that add to 5 are 2 and 3, so: $$x^2 + 5x + 6 = (x + 2)(x + 3)$$ 9. If factoring is not straightforward, use the quadratic formula to find roots. 10. This expression is fundamental in algebra and appears in many problems involving parabolas, optimization, and physics.