Expanding and Simplifying Algebraic Expressions

We are asked to expand and simplify the given algebraic expression: $\LaTeX{(x - 5)^2 + (x + 3)^2 + 4x}$.

To solve this, we need to expand the squared terms using standard algebraic identities and then combine like terms.

Using Algebraic Expansion Formulas

We will use the following two standard formulas for squaring binomials:

• Square of a difference: $\LaTeX{(a - b)^2 = a^2 - 2ab + b^2}$
• Square of a sum: $\LaTeX{(a + b)^2 = a^2 + 2ab + b^2}$

Step-by-Step Expansion and Simplification Process

Let's expand each part of the expression and then combine them.

Step 1: Expand $\LaTeX{(x - 5)^2}$

Using the formula $\LaTeX{(a - b)^2 = a^2 - 2ab + b^2}$ with $\LaTeX{a=x}$ and $\LaTeX{b=5}$:

$\LaTeX{(x - 5)^2 = x^2 - 2(x)(5) + 5^2}$

$\LaTeX{(x - 5)^2 = x^2 - 10x + 25}$

Step 2: Expand $\LaTeX{(x + 3)^2}$

Using the formula $\LaTeX{(a + b)^2 = a^2 + 2ab + b^2}$ with $\LaTeX{a=x}$ and $\LaTeX{b=3}$:

$\LaTeX{(x + 3)^2 = x^2 + 2(x)(3) + 3^2}$

$\LaTeX{(x + 3)^2 = x^2 + 6x + 9}$

Step 3: Combine the Expanded Forms with the Remaining Term

Now, substitute the results from Step 1 and Step 2 back into the original expression $\LaTeX{(x - 5)^2 + (x + 3)^2 + 4x}$:

Original Expression = $\LaTeX{(x^2 - 10x + 25) + (x^2 + 6x + 9) + 4x}$

Step 4: Combine Like Terms

Group terms that have the same variable and exponent:

• Combine $\LaTeX{x^2}$ terms: $\LaTeX{x^2 + x^2 = 2x^2}$
• Combine $\LaTeX{x}$ terms: $\LaTeX{-10x + 6x + 4x}$
• First combine $\LaTeX{-10x + 6x = -4x}$
• Then combine $\LaTeX{-4x + 4x = 0x = 0}$
• Combine constant terms: $\LaTeX{25 + 9 = 34}$

Putting these combined terms together:

Simplified Expression = $\LaTeX{2x^2 + 0 + 34}$

Simplified Expression = $\LaTeX{2x^2 + 34}$

Step 5: Factor the Resulting Expression

The expression $\LaTeX{2x^2 + 34}$ has a common factor of 2. We can factor out 2:

$\LaTeX{2x^2 + 34 = 2(x^2 + 17)}$

Final Simplified Algebraic Expression

The fully expanded and simplified expression is $\LaTeX{2(x^2 + 17)}$.

Let's check this against the given options:

Our final simplified expression $\LaTeX{2(x^2 + 17)}$ matches Option 2.

Revision Table: Key Steps for Algebraic Simplification

Additional Information: Importance of Algebraic Identities

Algebraic identities are equations that are true for all possible values of the variables involved. They are fundamental tools in algebra for expanding, factoring, and simplifying expressions quickly and accurately.

Mastering these identities, such as the square of a sum or difference, and the difference of squares, is crucial for solving more complex problems in algebra and calculus.

Practicing with different expressions helps in recognizing when and how to apply these identities effectively to simplify algebraic expressions.