Question

the trinomial $x^2 + bx + c$ factors to $(x + m)(x + n)$. if $b$ is negative and $c$ is positive, what must be true about $m$ and $n$?
$\bigcirc$ $m$ and $n$ are both positive.
$\bigcirc$ $m$ is positive and $n$ is negative.
$\bigcirc$ $m$ is negative and $n$ is positive.
$\bigcirc$ $m$ and $n$ are both negative.

Explanation:

Step1: Expand factored form

$$(x+m)(x+n) = x^2 + (m+n)x + mn$$

Step2: Match coefficients

Compare to $x^2 + bx + c$:
$b = m+n$, $c = mn$

Step3: Analyze sign conditions

Given $c>0$: $mn>0$, so $m,n$ same sign.
Given $b<0$: $m+n<0$, so sum is negative.

Step4: Deduce signs of m,n

Same sign + negative sum = both negative.

Answer:

m and n are both negative.