Solve the following multiplications using the distributive method: a) 28 x 43 b) 39 x 48 c) 46 x 57

We use the distributive method:

$$28 \times 43 = (20 + 8) \times (40 + 3)$$ $$= (20 \times 40) + (20 \times 3) + (8 \times 40) + (8 \times 3)$$ $$= 800 + 60 + 320 + 24$$ $$= 1180 + 24$$ $$= 1204$$

The student's calculation for $28 \times 43$ was:
$8 \times 3 = 24$
$20 \times 3 = 60$
$8 \times 40 = 320$
$20 \times 40 = 800$
Summing these: $24 + 60 + 320 + 800 = 1204$.
The student's final answer for a) is 1204.

We use the distributive method:

$$39 \times 48 = (30 + 9) \times (40 + 8)$$ $$= (30 \times 40) + (30 \times 8) + (9 \times 40) + (9 \times 8)$$ $$= 1200 + 240 + 360 + 72$$ $$= 1440 + 360 + 72$$ $$= 1800 + 72$$ $$= 1872$$

The student's calculation for b) is:
$9 \times 8 = 72$
$30 \times 8 = 240$
$9 \times 40 = 360$
$30 \times 40 = 1200$
Summing these: $72 + 240 + 360 + 1200 = 1872$.
The student's final answer for b) is 1872.

We use the distributive method:

$$46 \times 57 = (40 + 6) \times (50 + 7)$$ $$= (40 \times 50) + (40 \times 7) + (6 \times 50) + (6 \times 7)$$ $$= 2000 + 280 + 300 + 42$$ $$= 2280 + 300 + 42$$ $$= 2580 + 42$$ $$= 2622$$

The student's calculation for c) is:
$6 \times 7 = 42$
$40 \times 7 = 280$
$6 \times 50 = 300$
$40 \times 50 = 2000$
Summing these: $42 + 280 + 300 + 2000 = 2622$.
The student's final answer for c) is 2622.

The student's given answer is 136.
We use the distributive method:

$$67 \times 72 = (60 + 7) \times (70 + 2)$$ $$= (60 \times 70) + (60 \times 2) + (7 \times 70) + (7 \times 2)$$ $$= 4200 + 120 + 490 + 14$$ $$= 4320 + 490 + 14$$ $$= 4810 + 14$$ $$= 4824$$

The student's answer of 136 is ❌ Incorrect.
The correct answer is $\boxed{4824}$.

We use the distributive method:

$$84 \times 93 = (80 + 4) \times (90 + 3)$$ $$= (80 \times 90) + (80 \times 3) + (4 \times 90) + (4 \times 3)$$ $$= 7200 + 240 + 360 + 12$$ $$= 7440 + 360 + 12$$ $$= 7800 + 12$$ $$= 7812$$

The final answer is $\boxed{7812}$.