The most clue on the board is the Gray 9 in the bottom-left corner. Since 9 is a perfect square (3x3) and its clue is in a corner, its placement is forced. It must be a 3x3 square occupying that corner. This single move provides a massive, unambiguous starting point.

The rightmost column is our next target. The grid is 8 units high, and we have four clues along that edge: Teal 2, Purple 3, Pink 2, and the corner Brown piece. The only way to tile an 8-unit column with these shapes is to orient them vertically and find a containment. After a little trial, the unique combination becomes clear: Teal 2 must be a 1x2 rectangle (2 units high).

Purple 3 must be a 1x3 rectangle (3 units high).

Pink 2 must be a 2x1 rectangle (using 1 unit of height in that column).

This leaves 8 - (2+3+1) = 2 units of height for the Brown shape, forcing it to be a 2x2 square. This chain reaction solves the entire right border in one strategic swoop.

Now we return to the left side. The 3x3 Gray square takes up the bottom 3 rows. This leaves a 5-unit vertical space above it for the Green 4, Gold 2, and the corner Red piece. Similar to the right edge, we look for a containment. The solution is a stack of three shapes, all 2 units wide: A 2x2 Red square (height 2).