Question 403073
{{{21x^3-18x^2y+24xy^2}}}
When factoring, always start with the Greatest Common Factor (GCF). The GCF here is 3x:
{{{3x(7x^2-6xy+8y^2)}}}
Always keep factoring until no more factoring can be done. The second factor is a trinomial we should try to factor. It does not fit any of the patterns for factoring. But we can try trinomial factoring. This factoring is "un-FOIL-ing" the expression. We are looking to see if some pair of binomials will result in {{{7x^2-6xy+8y^2}}} when multiplied. We are looking for
{{{7x^2-6xy+8y^2 = (a[1] + b[1])(a[2] + b[2])}}}
where {{{a[1]}}} and {{{a[2]}}} (the "First" terms) are factors of {{{7x^2}}} and {{{b[1]}}} and {{{b[2]}}} (the "Last" terms) are factors of {{{8y^2}}}. That is the easy part the hard part is that the "Outside" terms, {{{a[1]}}} and {{{b[2]}}}, and "Inside" terms, {{{a[2]}}} and {{{b[1]}}} must produce the -6xy if the expression is to factor at all.<br>
For {{{a[1]}}} and {{{a[2]}}} there is really only one choice: 7x and x. For {{{b[1]}}} and {{{b[2]}}} there are several choices. Since the middle term has a minus in front of it we need them both to be negative. But they could be -8y and -y or -2y and -4y. And which one is {{{b[1]}}} and which one is {{{b[2]}}} is also not known yet.<br>
So the possibilities are:
(7x-8y)(x-y)
(7x-y)(x-8y)
(7x-2y)(x-4y)
or
(7x-4y)(x-2y)
All of these will produce the {{{7x^2}}} and the {{{8y^2}}}. But only 1 (or none) of these will produce the -6xy. And that will come from the Outside and Inside terms. Let's try each one, looking at just the Outside and Inside multiplications of FOIL:
(7x-8y)(x-y)
Outside: 7x*(-y) = -7xy
Inside: x*(-8y)  = -8xy
Together they make -15xy<br>
(7x-y)(x-8y)
Outside: 7x*(-8y) = -56xy
Inside: x*(-y)  = -xy
Together they make -57xy<br>
(7x-2y)(x-4y)
Outside: 7x*(-4y) = -28xy
Inside: x*(-2y)  = -2xy
Together they make -30xy<br>
(7x-4y)(x-2y)
Outside: 7x*(-2y) = -14xy
Inside: x*(-4y)  = -4xy
Together they make -18xy<br>
None of these produced the -6xy we were looking for. So {{{7x^2-6xy+8y^2}}} will not factor any further. So your fully factored expression is:
{{{3x(7x^2-6xy+8y^2)}}}