Question 1041380
{{{2x^2 + 4x + p=0}}} <==> {{{x^2 + 2x + p/2=0}}}

==> {{{-2 = alpha + beta}}} and {{{alpha*beta = p/2}}}.

Now, {{{x^2+4x+6 = 0}}} ==> {{{-4 = k/alpha + k/beta}}}

<==> {{{-4 = (k(alpha+beta))/(alpha*beta))}}}

==> {{{-4 = (k*-2)/(p/2)}}}  ==> k = p, after clearing fractions and simplifying.

Also, from the given, {{{6  =(k/alpha)*(k/beta) = k^2/(alpha*beta)}}}

==> {{{6*alpha*beta = k^2}}}  ==> {{{6*(p/2) = k^2}

==> {{{3p = k^2}}}   ===> {{{3p = p^2}}}  ===> p = 0,3.

p = 0 is not acceptable because it will imply a root of 0 in {{{2x^2 + 4x + p=0}}} and an undefined root in  {{{x^2+4x+6 = 0}}}.

Therefore, {{{highlight(k = p = 3)}}}.