Answer:
Infinity
Step-by-step explanation:
Since [tex]x^4 ax^3 3x^2 bx 1 \ge 0[/tex] for all real numbers x, this property is also true for x=1, which tells us that
[tex]1^4 a1^3 3\cdot1^2 b\cdot 1=3ab\ge0[/tex]
On the other hand, note that for all real numbers x, it holds that
[tex]x^4x^3x^2x\ge 0[/tex]
Therefore, if
[tex]3ab\geq0[/tex]
we have tat
[tex]3ab x^{4}x^{3}x^{2}x^{1}1=x^4ax^3 3x^2bx1\ge0[/tex]
The last reasoning tells us that the property [tex]x^4 ax^3 3x^2 bx 1 \ge 0[/tex] holds for all real numbers x if an only if [tex]ab\ge0[/tex]
Therefore, we can choose arbitrary constants a and b as long as
[tex]ab\ge0[/tex]
We can choose a and b such that both positive, both negative or one of the two constants is equal two zero. In the first two cases
[tex]a^2b^2[/tex]
can get as big as we want, depending on the constants, and we are done.
