Analyzing the Expression (x^3 + 3x^2  4x + 5) / (x^2  2)
This expression represents a rational function, where the numerator is a cubic polynomial (x^3 + 3x^2  4x + 5) and the denominator is a quadratic polynomial (x^2  2). Let's break down how to analyze and manipulate this expression.
1. Factoring
 Denominator: The denominator (x^2  2) can be factored using the difference of squares pattern: (x + √2)(x  √2). However, the numerator (x^3 + 3x^2  4x + 5) is not easily factorable by standard methods.
2. Long Division
Since the degree of the numerator (3) is greater than the degree of the denominator (2), we can use polynomial long division to simplify the expression:
x + 3
x^2  2  x^3 + 3x^2  4x + 5
(x^3  2x)

3x^2  2x + 5
(3x^2  6)

2x + 11
This gives us:
(x^3 + 3x^2  4x + 5) / (x^2  2) = x + 3 + (2x + 11) / (x^2  2)
3. Partial Fractions (Optional)
The remaining term (2x + 11) / (x^2  2) can be further decomposed using partial fraction decomposition. This technique involves expressing the rational function as a sum of simpler fractions. Since the denominator has distinct linear factors (x + √2) and (x  √2), the decomposition would look like this:
(2x + 11) / (x^2  2) = A / (x + √2) + B / (x  √2)
where A and B are constants. To find A and B, you would solve a system of equations.
4. Graphing
The simplified expression x + 3 + (2x + 11) / (x^2  2) allows for easier analysis and graphing.

Asymptotes: The vertical asymptotes occur at x = √2 and x = √2, where the denominator becomes zero. The oblique asymptote is y = x + 3, determined from the long division result.

Intercepts: The yintercept occurs when x = 0, giving a point (0, 5/(2)). To find the xintercepts, you would need to solve the equation x + 3 + (2x + 11) / (x^2  2) = 0, which is more complex.
5. Domain and Range
The domain of the function is all real numbers except for x = √2 and x = √2. The range is more complex to determine, requiring analysis of the function's behavior near the asymptotes and other critical points.
In Summary
By performing long division, we can express the given rational function as a linear term (x + 3) plus a simpler rational term. This allows for easier analysis of the function's behavior, including finding asymptotes and determining the domain. Depending on the specific application, further decomposition using partial fractions may be necessary.