When multiplying $(x+m)(x+n)$ using FOIL, which pair of terms combine to produce the $bx$ term in $x^2+bx+c$?

Multiply the binomials $(x-6)(x+2)$.

Factor $x^2+11x+24$.

A landscaper models the area (in units of square meters) of a rectangular flower bed as: $A\left(x\right)=x^2+5x+6$, where $x$ is a side-length adjustment in units of meters. Factor $A\left(x\right)$ to find expressions for the possible integer side lengths (in units of meters) and write the final results in terms of $x$.

Explain, using a short algebraic derivation starting from: $(x+m)(x+n)$, why $m+n$ must equal $b$ and $m\cdot n$ must equal $c$ in: $x^2+bx+c$. Which of the following derivations shown below is correct?