Finding area of rectangle under a parabola asymmetrical with respect to the Y-axis: What did I do wrong?

Question

I am using these as references:

1. How to find the dimensions of a rectangle if its area is to be a maximum?

2. Does the symmetry of a parabola in finding the maximum area of a rectangle under said parabola matter? (related)

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Problem: Find the maximum area of a rectangle inside a parabola whose equation is $f(x)=-x^2+4x+3$

My Work:

1. Determining the rough shape of the parabola (by determining the highest point of the graph which is $(2,7)$ and the y-intercept which is $y=3$), and placing the rough location of the rectangle inside the parabola.

2. Shifting the parabola to the left so that the $Y$ axis becomes the axis of the parabola (i.e. $f(x)=f(x+(−b/2a))$). The shift results with a new equation of the parabola, which is $f(x)=-x^2+7$. Graphing it.

3. From reference 2, the area's equation is $A(x)=2(x-(-b/2a))(-x^2+7)$ (I suspect this is where my mistake lies) where $-b/2a = 2$, the equation turns out to be a cubic equation $A(x)=-2x^3+4x^2+14x-28$
4. $A'(x)=0$ (to obtain the maximum $A(x)$), with roots $x=-2/3$ and $x=2$
5. Substitute $x=2$ to $A(x)$, $A(2)=-2(8)+4(4)+28-28$ which turns out to be $0$

Thank you for the help...

Answer 1

After you shift the parabola, you should use new parameters in $f(x)=-x^2+7$. That means $-b/2a=0$.

Answer 2

My mistake indeed lies on Step 3, where I either have to:

1. Use the original equation of the parabola (which is $f(x)=-x^2+4x+3$), $A(x)=2(x−(−b/2a))(−x^2+4x+3)$, or
2. Use the new equation of the parabola i.e. the equation you got after transforming/shifting the parabola, $A(x)=2x(−x^2+7)$

Both equations derives from the original equation $A(x)=2xy$ (from my first reference).

Instead of doing either steps, I used the new parabola's equation ($−x^2+7$) while still subtracting the axis shift ($(x−(−b/2a))$) despite the fact with my new equation the parabola is already symmetric.

Thanks to Justpassingby and ZQD for the help!