If $\displaystyle f(x)=x^4+2x^3+5x^2-16x-20$ , show that f(x) can be expressed in the form $\displaystyle (x^2+x+a)^2-4(x+b)^2$ .
I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?
$\displaystyle (x^2+x+a)^2-4(x+b)^2$ .
$\displaystyle =x^4+x^2 + a^2 + 2x^3 + 2ax + 2ax^2 -4x^2-4b^2 -8xb $
$\displaystyle
=x^4 + 2x^3+(2a-3)x^2 + (2a-8b)x +(a^2 -4b^2)
$
Comparing above eqn. with
$\displaystyle f(x)=x^4+2x^3+5x^2-16x-20$
$\displaystyle
2a-3 = 5
$
$\displaystyle
\implies a=4
$
$\displaystyle
2a-8b= -16
$
$\displaystyle
\implies b= 3
$
You can check by putting the value
$\displaystyle
a^2-4b^2 = -20
$
This is what you did
Hence you have shown that values of a and b are real , proving that statement is absolutely correct