A data analyst is factoring the expression $x^2 - 11x + 24$ to simplify a calculation. According to the rules for factoring trinomials where the middle term is negative and the last term is positive, what must be the signs of the two numbers used in the factors?

A small business owner uses the expression x^2 - 15x + 54 to model the break-even point for a new product line. When factoring this expression into the form (x + m)(x + n), both m and n must be ____ integers.

A warehouse manager uses the expression x^2 - 12x + 32 to determine optimal shelf spacing. To factor this expression into the form (x + m)(x + n), both m and n must be negative integers.

A retail supply chain analyst uses the quadratic expression $x^2 - 13x + 40$ to model fluctuating shipping costs. To factor this expression into the form $(x + m)(x + n)$, match each mathematical observation with the conclusion it provides about the signs of the integers $m$ and $n$.

A project manager is evaluating a resource allocation model represented by the quadratic expression $x^2 - 17x + 60$. To factor this trinomial into the form $(x + m)(x + n)$, arrange the logical steps used to determine the signs of the integers $m$ and $n$ in the correct sequence.

Storage Unit Layout Factoring

Workspace Layout Optimization

Production Cost Factoring Logic

A project manager is evaluating a resource allocation model represented by the quadratic expression $x^2 - 13x + 40$. To factor this trinomial into the form $(x + m)(x + n)$, which mathematical requirements must the integers $m$ and $n$ satisfy?

A production manager is factoring the expression $x^2 - 18x + 80$ to optimize assembly line timing. To factor this trinomial into the form $(x + m)(x + n)$, the manager recalls that both $m$ and $n$ must be negative integers. Which of the following conditions on the product and sum of these integers requires them to both be negative?

Factoring $y^2 - 11y + 28$

Factoring $y^2 - 16y + 63$

Factoring $t^2 - 11t + 28$

Factoring $a^2 - 11ab + 10b^2$