9514 1404 393
 384 cm²
Step-by-step explanation:
It is always a good idea to take a look at what you have and what you are being asked before you go launching into formulas. Then you can pick the formula(s) appropriate to your needs.
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Your pyramid has a square base, so four (4) triangular faces. The area being asked for is the total of the areas of the square base and those triangular faces.
The formula for the area of a square is ...
 A = s² . . . . . where s represents the side length
Here, the side of the square is 12 cm, so the area of the square base will be ...
 A = (12 cm)² = 144 cm²
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The formula for the area of a triangle is ...
 A = 1/2bh . . . . . where b is the base length and h is the altitude
In this problem, you're not given the altitude of the triangular face (also called the "slant height" of the triangle). Instead, you're given the height of the pyramid. So, you have to determine the slant height.
Consider the cross section you get when you cut the pyramid with a vertical plane through the peak and the middle of opposite sides. It will be an isosceles triangle with a base equal to the side length of the pyramid, and side lengths equal to the slant height of the pyramid face. See the attachment for an idea of what this looks like.
In order to determine the length of the side of that cross-section triangle, draw its altitude. That is the 8 cm height of the pyramid. Then use the Pythagorean theorem to find the hypotenuse of the right triangle with legs 6 cm and 8 cm:
 slant height = √(8² +6²) = √100 = 10 . . . . cm
Now, we can find the area of one of the four triangular faces:
 A = (1/2)(12 cm)(10 cm) = 60 cm²
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The surface area of the pyramid is then ...
 surface area = base area + 4 × triangular face area
 surface area = 144 cm² + 4 × 60 cm² = (144 +240) cm²
 surface area = 384 cm²
