Let us call L the length of the hall and w its width.
Then the area of the hall is

and we are told that is equal to 896 ft; therefore,

Now at this point, we have to remember that we are told that the width is 4 less than length; therefore, w = L -4 and the above equation becomes

Now we have to solve for L.
Expanding the left- hand side of the above equation gives

subtract 896 from both sides to get

Now the quadratic formula says that if we have an equation of the form

then
![L=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}](https://img.qammunity.org/2023/formulas/mathematics/college/6opb4fe1bjndqs2wey3yv77963ljzz0svk.png)
Now in our case we have L instead of L and b = -4, c = -896; therefore,
![L=\frac{-(-4)\pm\sqrt[]{(-4)^2-4(1)(-896)}}{2(1)}](https://img.qammunity.org/2023/formulas/mathematics/college/tbm4cwpt06qevbkdcq2pxtbom6eu8n28ut.png)
Simplifying the right - hand side of the above equation gives
![L=\frac{4\pm\sqrt[]{16+3584}}{2}](https://img.qammunity.org/2023/formulas/mathematics/college/f6htav9h1ch3emutvnmvznp6a5440hkmwg.png)
![x=(4)/(2)\pm\frac{\sqrt[]{16+3584}}{2}](https://img.qammunity.org/2023/formulas/mathematics/college/ifpw2hg29rkoh2ob6ely9c8e5tnunzp9vl.png)
which gives

Therefore the two values of L we get are


SInce lengths cannot be negative, the value L = 32 is the right answer.
Hence, the length of the hall is 32 ft.