This one is very challenging at your level unless you know what is going on. Knowledge of algebra at the secondary level makes solving this much easier.

Upon very close observation, for the black buttons, the number of black buttons in each figure is given by

(Fig 1) 0 x 1 div by 2
(Fig 2) 1 x 2 div by 2
(Fig 3) 2 x 3 div by 2
(Fig 4) 3 x 4 div by 2
(Fig 5) 4 x 5 div by 2
...
(Fig n) (n - 1) x n div by 2

As for the white buttons, it’s simply the square of the figure number, n^2.

The total sum of buttons is the sum of the two expressions above.

Since you have not learnt algebra in detail in Primary 6, I keep things simple with the algebraic lettering and we use mainly Guess and check here.

Figure n contains
(n - 1) x n / 2 black squares
n^2 white squares

For the figure number which contains 176 buttons, you can continue the pattern numbering from the list.

Pattern 6 contains 15 black buttons and 36 white buttons. Total is not 176 buttons.

Pattern 7 contains 21 black buttons and 49 white buttons. Total is not 176 buttons.

Pattern 8 contains 28 black buttons and 64 white buttons. Total is not 176 buttons.

Pattern 9 contains 36 black buttons and 81 white buttons. Total is not 176 buttons.

Pattern 10 contains 45 black buttons and 100 white buttons. Total is not 176 buttons.

Pattern 11 contains 55 black buttons and 121 white buttons. Total is 176 buttons.

For the second part, do not continue this way as it will take very Long to get to 784 white buttons.

Remember that the number of white buttons is

Figure number x Figure number.

We find two identical numbers which multiply to 784. Have you learnt square roots?

If not, you can do a Guess and check here. You will find that this is pattern 28, since 28 x 28 = 784.

Remember that the number of black buttons is given by

(n - 1) x n / 2.

Here the pattern number is 28, so n = 28.

We replace n by 28 to find the number of black buttons, which is equal to

27 x 28 / 2 = 378

Noted, Thank You so much for detailed explanation!