To find the solutions to the recurrence relation $a_n = 2a_{n-1} + 2n + 2$, we can use the method of solving non-homogeneous recurrence relations. Let’s go through the steps:

Step 1: Identify the Homogeneous Part

The homogeneous part of the recurrence relation is: $a_n^h = 2a_{n-1}^h$ This is a first-order linear recurrence relation with the solution of the form: $a_n^h = C \cdot 2^n$ where $C$ is a constant.

Step 2: Find a Particular Solution

Now we need a particular solution for the non-homogeneous equation: $a_n = 2a_{n-1} + 2n + 2$ The non-homogeneous part $2n + 2$ suggests a linear function as a particular solution. Let’s assume $a_n^p = An + B$ as our particular solution form, where $A$ and $B$ are constants to be determined.

Substitute $a_n^p = An + B$ into the recurrence relation: $An + B = 2(A(n-1) + B) + 2n + 2$ Expanding and simplifying, $An + B = 2An - 2A + 2B + 2n + 2$ Now, equate the coefficients of $n$ and the constant terms on both sides:

1. Coefficient of $n$: $A = 2A + 2 \Rightarrow -A = 2 \Rightarrow A = -2$
2. Constant term: ( B = -2A + 2B + 2 \Rightarrow B = -2(-2) + 2B + 2 \Rightarrow B = 4 + 2 - B \Rightarrow B