a$.$ To formulate the model algebraically, we first define the decision variables:

Let D represent the number of doors produced and W represent the number of windows produced.

The objective function is to maximize the total profit, which can be expressed as:

Maximize Z $=500$D $+350$W $-25$D$^2-66$W$^2$

Subject to the following constraints:

Plant $1$: D $<=1($availability of doors$),$ W $<=0($availability of windows$),$ and $4$D $+0$W $<=4($hours available$)$

Plant $2$: D $<=0($availability of doors$),$ W $<=2($availability of windows$),$ and $0$D $+2$W $<=12($hours available$)$

Plant $3$: D $<=3($availability of doors$),$ W $<=2($availability of windows$),$ and $3$D $+2$W $<=18($hours available$)$

b$.$ To solve the model on a spreadsheet using the solver:

Set up a spreadsheet with columns for D and W$,$ and a row for the objective function and constraints.

Input the coefficients for the objective function and constraints based on the given information.

Use the solver tool to maximize the objective function subject to the constraints, ensuring that the number of products is required to be integer.

The final optimal values for D and W obtained from the solver will provide the solution to the problem