Factoring $z^2 + 4z - 5$

Factoring $z^2 - 4z - 5$

Comparing the Factorizations of $z^2 + 4z - 5$ and $z^2 - 4z - 5$

Factoring $q^2 - 2q - 15$

A facility manager is calculating the dimensions of a rectangular storage area represented by the expression $x^2 + 2x - 15$. To factor this expression into the form $(x + m)(x + n)$, what must be true about the signs of the numbers m and n because the constant term (-15) is negative?

A logistics coordinator is modeling the floor space of a new warehouse using the expression $x^2 + 5x - 24$. To factor this into the form $(x + m)(x + n)$, the coordinator must identify two numbers, $m$ and $n$, that have ____ signs because their product is negative.

A logistics coordinator is using the expression x^2 + bx + c to model the net profit of a shipping route, where the constant term c represents a fixed loss (a negative value). Match each component of the factoring process for the form (x + m)(x + n) with its required rule.

An operations analyst is using the expression $x^2 + 4x - 12$ to model a production cycle, where the constant term -12 represents a resource deficit. To factor this expression into the form $(x + m)(x + n)$, the numbers $m$ and $n$ must have the same sign (both positive or both negative).

Sign Rule for Factoring with Negative Constants

A production analyst is using the quadratic expression $x^2 + bx + c$ to model resource allocation, where a negative constant term ($c$) represents a resource deficit. Arrange the following steps in the correct logical order to factor this expression into the form $(x + m)(x + n)$.

Resource Allocation Deficit Analysis

Sign Rules for Factoring Negative Deficits

A supply chain analyst is using the expression $x^2 + bx + c$ to model quarterly inventory shifts, where the constant term $c$ is negative. When factoring this into the form $(x + m)(x + n)$, what determines whether the middle coefficient $b$ is positive or negative?

A logistics coordinator is using the expression $x^2 + bx + c$ to model inventory fluctuations, where a negative constant term $c$ represents a stock deficit. When factoring this expression into the form $(x + m)(x + n)$, which fundamental rule explains why $m$ and $n$ must have opposite signs?