Question 1164354
<br>
It is almost certain that you have not shown the expressions correctly.<br>
Specifically, you show two linear y terms and no linear x term.<br>
It would seem advantageous to you to check your post before you submit it....<br>
Nevertheless, I will go through this with you, because it is an interesting problem.<br>
I have changed the "-8y" term to "-8x"....<br>
{{{6x^2+7xy-3y^2-8x+10y+c = (Ax+By+C)(Dx+Ey+F)}}}<br>
We know the coefficient of the x^2 term is AD=6; we need to find the constant term CF.<br>
Since AD is 6, we can have A and D be either 6 and 1, or 3 and 2.  It is more likely that they are 3 and 2; so let's try that and see if we can reach an answer.<br>
{{{(3x+By+C)(2x+Ey+F) = 6x^2 + (2B+3E)xy + (BE)y^2 + (2C+3F)x + (BF+CE)y + CF}}}<br>
Equating coefficients with the given expression, we have<br>
{{{2B+3E = 7}}}
{{{BE = -3}}}
{{{2C+3F = -8}}}
{{{BF+CE = 10}}}<br>
Without going through the details (they aren't hard), the first two equations give us B = -1 and E = 3.<br>
Then, knowing B = -1 and E = 3, the last two equations give us<br>
{{{2C+3F = -8}}}
{{{3C-F = 10}}}<br>
Again without the details, those two equations give us C = 2 and F = -4.<br>
And that gives us our answer: CF = -8.<br>
The result is easily confirmed by doing the multiplication:<br>
{{{(3x-y+2)(2x+3y-4) = (6x^2+9xy-12x)+(-2xy-3y^2+4y)+(4x+6y-8) = 6x^2+7xy-3y^2-8x+10y-8}}}<br>