The Quadratic polynomials defined on real coefficients $\mathrm{P}\left(\mathrm{x}\right)={\mathrm{a}}_1{\mathrm{x}}^2+2{\mathrm{b}}_1\mathrm{x}+{\mathrm{c}}_1;\mathrm{Q}\left(\mathrm{x}\right)={\mathrm{a}}_2{\mathrm{x}}^2+2{\mathrm{b}}_2\mathrm{x}+{\mathrm{c}}_2$; where a₁ $\ne$0,a₂ $\ne$ 0 and P(x) and Q(x) both take positive values $\forall$x $\in$R . g(x) = a₁a₂x ²+b₁b₂x + c₁c₂ then

The Quadratic polynomials defined on real coefficients $\mathrm{P}\left(\mathrm{x}\right)={\mathrm{a}}_1{\mathrm{x}}^2+2{\mathrm{b}}_1\mathrm{x}+{\mathrm{c}}_1;\mathrm{Q}\left(\mathrm{x}\right)={\mathrm{a}}_2{\mathrm{x}}^2+2{\mathrm{b}}_2\mathrm{x}+{\mathrm{c}}_2$;take positive values $\forall$x $\in$R 

$\Rightarrow$D₁ = 4b²₁ -4a₁c₁ <0, D₂ = 4b₂²  -4a₂c₂ < 0
$\Rightarrow$a₁a₂c₁c_{2 >} b₁²b₂² then  For the equation g(x) = a₁a₂x ²+b₁b₂x + c₁c₂

D₃ =(b₁b₂)²   - 4a₁a₂c₁c₂ <0

$\Rightarrow$g(x) takes +ve values only  $\forall$x $\in$R