Compute the product $(x+4)(x-5)$.

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?

You encounter the expression: $3x^2+14x+8$. A peer suggests that directly finding integers which will add to 14 and multiply to 8 will factor this expression. Is this suggestion valid? Choose the best justification.

A design engineer examines the quadratic expression: $P\left(x\right)=x^2+\left(k\right)x+41$, where $k$ is an integer. They want to solve for the integer linear factors, but they must note that $41$ is prime. For which integer value(s) of $k$ can $x^2+kx+41$ be factored into integers $(x+m)(x+n)$?