polynomial

Let a and b real numbers such that   \(x^4+2x^3-4x^2+ax+b = (Q(x))^2\) for some polynomial Q(x).

What is the value of a + b?

 

\(x^4+2x^3-4x^2+ax+b = (Q(x))^2\\ (Q(x))^2=(x^2+mx+t)(x^2+mx+t) \\ (Q(x))^2=x^4+mx^3+tx^2\;\;+mx^3+m^2x^2+mtx\;+\;tx^2+mtx+t^2\\ (Q(x))^2=x^4+2mx^3+2tx^2\;\;+m^2x^2+2mtx+t^2\\ (Q(x))^2=x^4+2mx^3+(2t+m^2)x^2+2mtx+t^2\\\)
\(2m=2\;  \;so\;\; m=1\\ 2t+m^2=-4\\ 2t+1=-4\\ t=-2.5\\~\\ 2mt=a\\ 2*1*-2.5=a\\ a=-5\\~\\ t^2=b\\ b=(-2.5)^2\\ b=6.25\\~\\ a+b=6.25-5=1.25 \)

 

 

 

 

 

Latex

x^4+2x^3-4x^2+ax+b = (Q(x))^2\\
(Q(x))^2=(x^2+mx+t)(x^2+mx+t) \\
(Q(x))^2=x^4+mx^3+tx^2\;\;+mx^3+m^2x^2+mtx\;+\;tx^2+mtx+t^2\\
(Q(x))^2=x^4+2mx^3+2tx^2\;\;+m^2x^2+2mtx+t^2\\
(Q(x))^2=x^4+2mx^3+(2t+m^2)x^2+2mtx+t^2\\