Question

factor the trinomial.
$v^{2}-vx - 42x^{2}$
$v^{2}-vx - 42x^{2}=\square$

Explanation:

Step1: Identify target product and sum

We need two terms that multiply to $-42v^2x^2$ (product of first and last term of trinomial) and add to $-vx$ (middle term). These terms are $6vx$ and $-7vx$.

Step2: Split the middle term

Rewrite the trinomial by splitting the middle term using the two identified terms:
$v^2 + 6vx - 7vx - 42x^2$

Step3: Factor by grouping

Group the first two and last two terms, then factor out the greatest common factor from each group:
$v(v + 6x) - 7x(v + 6x)$

Step4: Factor out common binomial

Factor out the shared binomial factor $(v + 6x)$:
$(v + 6x)(v - 7x)$

Answer:

$(v + 6x)(v - 7x)$