Can someone help explain how i would solve this using the pascals triangle? Like how to convert it into (x + 4)^3 = ax^b+cx^d+ex^f+g

Pascal's triangle is a way of organizing the coefficients in the binomial expansion, :

n = 0   =>  1

n = 1    =>  1   1

n = 2   =>  1   2   1

n = 3   =>  1   3   3   1

n = 4   =>  1   4   6   4   1

and so on, where each row starts and ends with a 1, and the numbers in the middle are obtained by adding together the two numbers directly above and to the left. These coefficients are then mulitplied by descending powers of a and descending powers of b (or vice versa, because the pattern is symmetric), starting with 0 and going up to n.

In this case, we have

Notice how the powers of x and 4 always sum to n = 3.

Simplifying, we get

so that

a = 1

b = 3

c = 12

d = 2

e = 48

f = 1

g = 64