Simplify (x^2-2x+2)/(x+4)+(2x-18)/(x+4)

Solution:

Step 1: Simplify the expression

Step 1.1: Combine the fractions over a common denominator

Combine the numerators since they share the same denominator: $\frac{x^2 - 2x + 2 + 2x - 18}{x + 4}$

Step 1.2: Simplify the numerator by combining like terms

Step 1.2.1: Cancel out the terms involving $x$

The terms $-2x$ and $2x$ cancel each other out: $\frac{x^2 + (0) + 2 - 18}{x + 4}$

Step 1.2.2: Combine the constant terms

Combine the constants in the numerator: $\frac{x^2 + (2 - 18)}{x + 4}$

Step 1.3: Subtract the constants

Subtract $18$ from $2$ to simplify further: $\frac{x^2 - 16}{x + 4}$

Step 2: Factor the numerator

Step 2.1: Express $16$ as a square of $4$

Rewrite the constant $16$ as $4^2$: $\frac{x^2 - 4^2}{x + 4}$

Step 2.2: Factor using the difference of squares

Apply the difference of squares formula: $\frac{(x + 4)(x - 4)}{x + 4}$

Step 3: Reduce the fraction

Step 3.1: Eliminate the common factor

Cancel out the common $(x + 4)$ term: $\frac{\cancel{(x + 4)}(x - 4)}{\cancel{(x + 4)}}$

Step 3.2: Simplify the result

The result simplifies to: $x - 4$

Knowledge Notes:

To solve the given problem, we applied the following knowledge points:

1. Combining Fractions: Fractions with the same denominator can be combined by adding their numerators.

2. Simplifying Expressions: This involves combining like terms and simplifying the expression step by step.

3. Difference of Squares: This is a special factoring technique where $a^2 - b^2$ can be factored into $(a + b)(a - b)$.

4. Reducing Fractions: If a term appears in both the numerator and the denominator, it can be canceled out, provided it's not equal to zero.

5. LaTeX Formatting: Mathematical expressions are rendered using LaTeX to provide clear and precise representations of equations and formulas.

By applying these principles, we were able to simplify the given algebraic expression step by step.