What is your question?
\(4+5x-6x^2 = (-43/4)H_1+5H_2 +(19/4)H_3\)
ok so we equate them and thats it?
that is the linear combination
read the question again
you want to check if there exist numbers \(a,b,c\) such that : \(4+5x-6x^2 = a*H_1+b*H_2 +c*H_3\)
plugin the given polynomials for \(H_1, H_2, H_3\) above and get : \(4+5x-6x^2 = a*x^2+b*(x-3) +c*(x^2+4)\)
compare coefficients both sides
maybe start by comparing \(x\) coefficient both sides
Look at below equation \(4+5x-6x^2 = a*x^2+b*(x-3) +c*(x^2+4)\)
whats the coefficient of x on left hand side ?
whats the coefficient of x on right hand side ?
Yes, the coefficients must be equal both sides, therefore \(b=5\)
next compare constant terms both sides and try finding another variable
whats the constant term on left hand side ?
what about right hand side ?
nope, try again
Yes, the constant terms must be equal on both sides : 4 = -3b + 4c
plugin the value of \(b\) and solve \(c\)
4=-3(5)+4c , 19=4c , 19/4=c
Yes, compare x^2 coefficients both sides and try finding the value of \(a\) too
i got a = -43/4
please double check ur work