Q.If p is real, discuss the nature of the roots of the equation $4x^2+4px+p+2=0$, in terms of p.

Computes the discriminant of the quadratic as a function of $p$, factorises it, and reads off the sign in each interval of $p$.

1. The equation is $4x^2+4px+(p+2)=0$, of the form $Ax^2+Bx+C=0$ with $A=4$, $B=4p$, $C=p+2$.
2. Discriminant: $D=B^2-4AC=(4p)^2-4(4)(p+2)=16p^2-16(p+2)$.
3. Simplify: $D=16\left[p^2-(p+2)\right]=16(p^2-p-2)$.
4. Factorise: $p^2-p-2=(p-2)(p+1)$, so $D=16(p-2)(p+1)$.
5. Since $16>0$, the sign of $D$ matches the sign of $(p-2)(p+1)$, which is a upward parabola in $p$ with roots $p=-1,2$.
6. Case $D>0$: $(p-2)(p+1)>0 \Rightarrow p<-1$ or $p>2$. Roots are real and distinct (unequal). …