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Algebraic expressions are combinations of numbers, variables, and operations. These expressions, unlike equations, do not have an equal sign. Variables are often represented by letters such as \( x \) and \( y \).
They play an essential part in mathematics, as they allow us to model real-world situations. When we evaluate an algebraic expression, we replace the variables with specific values. For example, in our exercise, the expression \( x^{2} + 2xy + y^{2} \) is evaluated with \( x=2 \) and \( y=-3 \). This step is crucial for simplifying and solving the expression to find its value in a particular scenario.
Understanding algebraic expressions helps you manipulate variables to find unknowns and tailor answers applicable to specific conditions.

The substitution method involves replacing variables with given numerical values. This is the first step in simplifying an expression. For example, if you have \( x=2 \) and \( y=-3 \), you can substitute these into your polynomial.

• Replace \( x \) with 2: so \( x^{2} \) becomes \( (2)^{2} \).
• Replace \( y \) with -3: so \( y^{2} \) becomes \( (-3)^{2} \) and \( 2xy \) becomes \( 2 \times 2 \times -3 \).

This method transforms an abstract expression into a numerical calculation, making it easier to understand and evaluate.
It is particularly helpful when you need to assess the expression's behavior under specific conditions, just like in real-world problems solving.

The simplification process helps in combining and reducing expressions to their most basic form. Simplification makes it easier to understand and calculate expressions.
In the given exercise, simplifying involves calculating powers, followed by addition and subtraction:

• First, calculate the powers: \( (2)^2 = 4 \), and \( (-3)^2 = 9 \).
• Second, handle the multiplication: \( 2 \times 2 \times -3 = -12 \).
• Finally, combine all terms to simplify: add and subtract step-by-step to get \( 4 - 12 + 9 \).

The simplification process requires careful handling of operations to ensure accuracy. When correctly simplified, the expression ultimately equals 1, which is the final result of the problem.
By practicing these skills, you'll improve your ability to tackle polynomial expressions efficiently.