Each shaded part on top as a fraction of the whole square : ½ × 1/5 = 1/10
Each shaded part below as a fraction of the whole square : ½ × ¼ = 1/8
Fraction of entire square that is shaded :
3 × 1/10 + 2 × 1/8
= 3/10 + 1/4
= 6/20 + 5/20
= 11/20
So, 11/20 of the square area = 165cm²
Area of square = 165cm² × 20/11 = 300cm²

Alternative working :
Top half and bottom half have the same area.
Top half is divided into 5 equal parts. Bottom half is divided into 4 equal parts.
Equal concept : Make the number of units the same for top and bottom.
Top half : 5 × 4u = 20u
Bottom half : 4 × 5u = 20u (lowest common multiple)
One shaded part on top = 4u, one shaded part below = 5u
Total shaded area = 3 × 4u + 2 × 5u = 12u + 10u = 22u
So 22u = 165cm²
1u = 165cm² ÷ 22 = 7.5cm²
Area of square = 20u + 20u = 40u
= 40 × 7.5cm²
= 300cm²