Question 319121
{{{8w^2-14w+3=(aw-b)(cw-d)}}}
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When you multiply the right hand side (FOIL method), you get,
{{{(aw+b)(cw+d)=acw^2+adw+bcw+bd}}}
{{{(aw-b)(cw-d)=(ac)w^2+(ad+bc)w+bd}}}
So you need to choose integers a,b,c,d so that,
{{{ac=8}}}
{{{ad+bc=-14}}}
{{{bd=3}}}
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As you have seen, when {{{b=-1}}}, {{{d=-3}}}, then 
{{{bd=3}}}
{{{-3a-c=-14}}}
which can be solved with 
{{{a=4}}} and {{{b=2}}}
Since {{{ab=8}}} so,
{{{8w^2-14w+3=(4w-1)(2w-3)}}}
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Best way to become proficient in this reverse FOIL method is to work plenty of forward FOIL problems to start to see the connections between the coefficients in the polynomials and the factors. 
Then the reverse process won't seem so daunting.