Very good so far.

Except that 3/2w is misleading. (3/2)w is better.

So is (3/2)w^2.

Now that you have your quadratic equation

A = 376w -(3/2)w^2

You transform that into

A = a(w-b) +c

so that you can get the vertex (b,c),

where b is the value of w at the vertex.

and c is the value of A at the vertex.

A = 376w -(3/2)w^2

A = -(3/2)[w^2 -(376w)/(3/2)]

A = -(3/2)[w^2 -(376*2 /3)w +(376/3)^2] -(3/2)(-(376/3)^2)

. Why don't we use decimal points?

A = -(1.5)[(w -125.333)^2] +23,562.667 ----------(i)

Parabola (i) is vertical, that opens downward because of the negative coefficient of w^2.

Therefore the vertex is the highest point, a maximum A.

Hence, for maximum A,

w = 125.333 m. ------------------------------answer.

L = 376 -(3/2)(125.333) = 188 m. -------------answer.

A = 23,562.7 sq.m. ---------------------------answer.

Check,

A = w*L

max A = 125.333 *188

23,562.7 =? 125.333 *188

23,562.7 =? 23,562.6

Umm, the 0.1 sq.m. difference in area was stolen by the neighbor.