To expand the expression (2+x)^5(1-2x)^6 as far as the term in x^3, we need to perform a binomial expansion using the binomial theorem. The binomial theorem states that for any two binomials (a + b)^n and (c + d)^m, the expanded form can be expressed as a sum of terms, where each term is obtained by multiplying one term from the first binomial with one term from the second binomial.

Let’s start with expanding (2+x)^5:

(2+x)^5 = C(5,0)*(2)^5*(x)^0 + C(5,1)*(2)^4*(x)^1 + C(5,2)*(2)^3*(x)^2 + C(5,3)*(2)^2*(x)^3 + …

C(n,k) represents the binomial coefficient, also known as “n choose k”, which calculates the number of ways to choose k items from a set of n items. In this case, C(5,0) = 1, C(5,1) = 5, C(5,2) = 10, C(5,3) = 10, and the terms continue in the same pattern.

Next, we expand (1-2x)^6:

(1-2x)^6 = C(6,0)*(1)^6*(-2x)^0 + C(6,1)*(1)^5*(-2x)^1 + C(6,2)*(1)^4*(-2x)^2 + C(6,3)*(1)^3*(-2x)^3 + …

Now, let’s combine the two expansions:

(2+x)^5(1-2x)^6 = [(2+x)^5]*[(1-2x)^6]

= [C(5,0)*(2)^5*(x)^0 + C(5,1)*(2)^4*(x)^1 + C(5,2)*(2)^3*(x)^2 + C(5,3)*(2)^2*(x)^3 + …]

- [C(6,0)
*(1)^6*(-2x)^0 + C(6,1)*(1)^5*(-2x)^1 + C(6,2)*(1)^4*(-2x)^2 + C(6,3)*(1)^3*(-2x)^3 + …]

To find the term involving x^3, we need to multiply terms from both expansions where the exponents of x add up to 3.

The term involving x^3 is obtained by multiplying the term C(5,1)*(2)^4*(x)^1 from the first expansion with the term C(6,2)*(1)^4*(-2x)^2 from the second expansion:

Term in x^3 = C(5,1)*C(6,2)*(2)^4*(x)^1*(1)^4*(-2x)^2

Simplifying this expression gives the term involving x^3 in the expansion of (2+x)^5(1-2x)^6.