The most logical starting point is the Green 7. As a prime number, its only possible dimensions are 1x7 or 7x1. Its clue is tucked into the top-right corner. A horizontal 1x7 strip is immediately impossible as it would run off the grid. This forces it to be a 7x1 vertical strip, locking down the entire rightmost column for seven of the eight rows. This single move provides a hard boundary for multiple other shapes.

With the Green 7 in place, the Yellow 4 is now constrained. Its clue is in the second-to-last column. It cannot be a 1x4 or a 2x2 shape without colliding with its green neighbor. The only remaining option is a 4x1 vertical strip, which fits next to the Green 7.

The Red 5 (another prime) can only be a 1x5 or 5x1. A 5x1 vertical strip is blocked by the Yellow 4 below it. This forces it into a 1x5 horizontal configuration.

The placement of the Red 5 immediately boxes in the Teal 4. It can't go up, right, or expand into a 2x2 or 4x1. It is squeezed into a 1x4 horizontal strip. The same logic then applies to the Light Blue 5, which is forced into a 1x5 horizontal shape by the pieces above and to its right.

With the majority of the board solved, the remaining pieces fall into place. The Purple 4 in the bottom-left corner has only one place to go: a 4x1 vertical strip in the first column. This final placement reveals the dimensions of the last two shapes. The empty space at the bottom must be the Brown patch (a 2x5 rectangle of area 10), and the remaining space at the top must be the Orange patch (also a 2x5 rectangle of area 10).