Recover parameters and check the complete curve

The Teacher completes the Student working with

$x^2-b=(x-\sqrt b)(x+\sqrt b),\qquad\sqrt b=2\Rightarrow b=4.$

Independently, a genuine vertical asymptote at 2 requires the original denominator to vanish there, so $4-b=0$. After finding $a$ below, also check the numerator at both denominator zeros to rule out removable holes.

With $b=4$, the source division has leading term $a$. Multiplying back gives $a(x^2-4)=ax^2-4a$; subtracting from $ax^2+4x+a$ leaves $4x+5a$. Hence

$y=a+\frac{4x+5a}{x^2-4}.$

The remainder tends to 0 at either infinite end, giving horizontal asymptote $y=a$. The given $y=3$ therefore gives $a=3$. This retains the Teacher’s full subtraction/remainder argument, not just its result.

Substitution gives $y=(3x^2+4x+3)/(x^2-4)$. Both source sketches label $x=-2,x=2,y=3$ and $(0,-3/4)$. Independently the numerator is 7 at −2 and 23 at 2, so both vertical asymptotes are genuine. Its discriminant is $4^2-4(3)(3)=-20<0$, hence there is no $x$-intercept. The $y$-intercept is $(0,-3/4)$.

Both source sketches label only the central local maximum $(-0.578,-0.461)$. Independent differentiation finds two stationary points over the full domain:

$\begin{aligned}&y'\\ & =\frac{(6x+4)(x^2-4)-2x(3x^2+4x+3)}{(x^2-4)^2}\\ & =-\frac{2(2x^2+15x+8)}{(x^2-4)^2}.\end{aligned}$

The roots are $x=(-15\pm\sqrt{161})/4$. The far-left root gives a local minimum near $(-6.922,2.711)$; the other gives the source-labelled maximum near $(-0.578,-0.461)$. Exact ordinates are respectively $(9+\sqrt{161})/8$ and $(9-\sqrt{161})/8$. Derivative signs are $-\to+$ at the first root and $+\to-$ at the second, splitting at the poles as required.

The source drawing does not display the far-left turn or crossing of the horizontal asymptote. Independently $y-3=(4x+15)/(x^2-4)$ vanishes at $x=-15/4$ and tends to 0 from below as $x\to-\infty$. A horizontal asymptote may be crossed. The complete reconstruction includes these features; the original source excerpt and its labelled maximum are preserved in review evidence.