The most decisive starting point is the Grey 4 in the bottom-right corner. While a 4-cell shape can be a 1x4 or 4x1 rectangle, its potential to be a 2x2 square is a constraint. Placing it as a 2x2 square in that corner immediately locks down a predictable, stable block of the grid.

With the Grey 4 set, the Gold shape above it is now trapped. It's bounded by the top of the Grey 4, the top of the grid, and the right edge of the grid. This leaves a space that is exactly 4 cells tall and 2 cells wide. This forces the Gold shape to be a 4x2 rectangle, revealing its area as 8.

The combination of the Gold 8 and the Grey 4 creates a solid "wall" six cells high along the right side. This new boundary becomes the key to solving the middle of the puzzle.

The Red 4 is now pushed up against this wall. The only way for a 4-cell shape to fit in the column next to the Gold/Grey wall is as a vertical 4x1 strip. No other configuration is possible.

The placement of the Red 4 now defines the right-hand boundary for both the Green 6 and the Purple 6. The Green 6 must be a 2x3 or 3x2 rectangle. A 2x3 rectangle fits in the space available. This, in turn, defines the bottom edge for the Purple 6, which must also be a 2x3 rectangle to fill the remaining space.

With the main blocks in place, the rest is simple. The Teal shape at the bottom is left with a 1x3 space, making it a 3. Likewise, the Light Blue 3 at the top is forced into the remaining 1x3 slot. The final piece, the Orange shape, fills the last 2x1 gap, revealing its value to be 2.