Question

factor the following trinomial, or state that the trinomial is prime.
$6x^{2}-23x+21$
select the correct choice below, and if necessary, fill in the answer box
a. $6x^{2}-23x+21=\square$
b. the trinomial is prime.

Explanation:

Step1: Multiply leading & constant terms

$6 \times 21 = 126$

Step2: Find pair summing to -23

We need two negative factors of 126 that add to -23: $-14$ and $-9$, since $(-14) \times (-9)=126$ and $-14 + (-9)=-23$

Step3: Split middle term

$6x^2 -14x -9x +21$

Step4: Factor by grouping

Group first/second and third/fourth terms:
$(6x^2 -14x) + (-9x +21)$
Factor out GCF from each group:
$2x(3x -7) -3(3x -7)$

Step5: Factor out common binomial

$(2x -3)(3x -7)$

Answer:

A. $6x^2 - 23x + 21 = (2x-3)(3x-7)$