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In the context of algebra, an \textbf{algebraic expression} is a combination of numbers, variables, and arithmetic operations. In an algebraic expression, variables represent unknown values and are often denoted by letters such as x, y, and z. Operations can include addition, subtraction, multiplication, division, and exponentiation.

For example, \( (y+1)^3 + 1 \) is an algebraic expression where \( y \) is the variable, 1 is a constant, addition is the operation, and the cube \( ^3 \) indicates exponentiation. Understanding how to manipulate these expressions is core to many algebraic concepts, including factoring polynomials and applying formulas like the sum of cubes.

Factoring is the process of breaking down a \textbf{polynomial} into simpler components, or 'factors,' that, when multiplied together, give the original polynomial. A polynomial is an algebraic expression with multiple terms—such as \( y^2 + 2y + 1 \)—typically involving the same variable raised to different powers.

When factoring polynomials, our goal is to find an equivalent expression that is easier to work with, often for the purposes of solving equations or simplifying complex expressions. The factored form can reveal information about the polynomial, such as its zeros or roots, which are the values of the variable that make the polynomial equal to zero.

For instance, in our exercise, we factor the sum of cubes in the expression \( (y+1)^3 + 1 \) to simplify it and find its factored form. This method is especially useful when dealing with higher degree polynomials where direct methods such as the quadratic formula cannot be applied.

The \textbf{sum of cubes formula} is a specialized factoring formula that applies to expressions where two terms are each raised to the third power and added together. The formula is \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \).

This formula is very powerful because it allows us to factor an expression that at first glance does not seem factorable. When applying the sum of cubes formula, we identify 'a' and 'b' from the expression \( a^3 + b^3 \). In the exercise we're looking at, \( a \) is \( (y+1) \) and \( b \) is \( 1 \).

As detailed in the step by step solution, after applying the formula to the given expression \( (y+1)^3 + 1 \), we can simplify to reach a factored form. This demonstrates the usefulness of the sum of cubes formula in transforming a seemingly non-factorable cubic expression into a factored expression that can be used for further algebraic operations.