Question 1050486
Use the discriminant,
{{{4x^2+12x+15=4px+5p}}}
{{{4x^2+12x-4px+15-5p=0}}}
{{{4x^2+(12-4p)x+(15-5p)=0}}}
So then,
{{{D=b^2-4ac}}}
{{{D=(12-4p)^2-4(15-5p)(4)}}}
For real roots,
{{{D>=0}}}
{{{ (12-4p)^2-16(15-5p)>=0 }}}
{{{(144-96p+16p^2)-240+80p>=0}}}
{{{16p^2-16p-96>0}}}
{{{p^2-p-6>=0}}}
{{{(p-3)(p+2)>=0}}}
Use the critical points, {{{p=3}}} and {{{p=-2}}},
Break up the number line into 3 regions,
Region 1 : {{{p<-2}}}
Region 2 : {{{-2<p<3}}}
Region 3 : {{{p>3}}}
Choose a value in each region and check the inequality,
Region 1 : {{{p=-3}}}
{{{(-3-3)(-3+2)>=0}}}
{{{-6(-1)>=0}}}
{{{6>=0}}}
True
Region 2 : {{{p=0}}}
{{{(0-3)(0+2)>=0}}}
{{{-6>=0}}}
False
Region 3 : {{{p=4}}}
{{{(4-3)(4+2)>=0}}}
{{{1(6)>=0}}}
{{{6>=0}}}
True
So then,
{{{p<=-2}}}U{{{p>=3}}}