(Use Pascal's triangle to expand each binomial. (x – 5y)^5

The n-th row in Pascal's triangle tells you the coefficients of terms in the expansion of (a + b)ⁿ. Starting with n = 0,

1

1 … 1

1 … 2 … 1

1 … 3 … 3 … 1

1 … 4 … 6 … 4 … 1

1 … 5 … 10 … 10 … 5 … 1

In more concrete terms, this translates to

(x - 5y)⁵ = 1 x ⁵ (-5y)⁰ + 5 x ⁴ (-5y)¹ + 10 x ³ (-5y)² + 10 x ² (-5y)³ + 5 x ¹ (-5y)⁴ + 1 x ⁰ (-5y)⁵

Simplify:

(x - 5y)⁵ = x ⁵ - 25x ⁴y + 250x ³y ² - 1250x ²y ³ + 3125xy ⁴ - 3125y ⁵

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!