Factoring $x^2 + bx + c$ When $b$ and $c$ Are Positive

A cost estimator for a manufacturing company uses the expression $x^2 - 15x + 54$ to model fluctuating utility expenses. To simplify the budget report, the estimator must factor this expression into two binomials of the form $(x + m)(x + n)$. According to the strategy for factoring trinomials where the constant term is positive and the middle coefficient is negative, what must be true about the signs of the integers $m$ and $n$?

A supply chain analyst is using a profitability model defined by the expression $x^2 + 8x - 20$ to determine inventory break-even points. To factor this trinomial into the form $(x + m)(x + n)$, the analyst notes that the constant term is negative and the middle coefficient is positive. According to the strategy for factoring trinomials, the factors $m$ and $n$ will have opposite signs, and the sign of the factor with the larger absolute value must be ____.

A logistics operations analyst uses quadratic expressions of the form $x^2 + bx + c$ to model cargo volume fluctuations. To simplify these models, the analyst must recall the sign rules for factoring these trinomials into the form $(x + m)(x + n)$. Match each set of conditions for the coefficients $b$ and $c$ with the corresponding rule for the signs of the factors $m$ and $n$.

Primary Indicator for Factor Signs

Factoring Signs in Sales Variance Analysis

An operations planner at a logistics firm uses the quadratic expression $x^2 - 12x + 32$ to model daily warehouse capacity fluctuations. To factor this expression into two binomials of the form $(x + m)(x + n)$, the planner applies standard factoring sign rules.

True or False: Because the constant term is positive (+32) and the middle coefficient is negative (-12), both of the factoring integers, $m$ and $n$, must be negative.