The given function in the image is:

$f(x) = 2 \cdot (x + i\sqrt{2})(x - i\sqrt{2})(x + \sqrt{5})(x - \sqrt{5})(x - 1)(x + 3)^2(x + 4)^3$

This expression includes both real and complex roots. Let me expand and analyze it in detail.

1. Simplification of the Factors

Complex Conjugate Pairs:

• $(x + i\sqrt{2})(x - i\sqrt{2}) = x^2 + 2$
• $(x + \sqrt{5})(x - \sqrt{5}) = x^2 - 5$

Real Terms:

• $x - 1$ remains as is.
• $(x + 3)^2$ remains as is.
• $(x + 4)^3$ remains as is.

2. Fully Expanded Form:

Now, substituting the simplified terms, the function becomes: $f(x) = 2 \cdot (x^2 + 2) \cdot (x^2 - 5) \cdot (x - 1) \cdot (x + 3)^2 \cdot (x + 4)^3$

To further expand, we can multiply these terms step-by-step.

Would you like me to expand the function completely, or analyze it for specific properties (like roots, degree, leading coefficient, etc.)? Let me know!

Related Questions:

1. How do we calculate the degree of this polynomial?
2. What are the roots of this polynomial, including their multiplicities?
3. How do complex conjugate roots simplify in a polynomial?
4. What is the leading coefficient of the expanded polynomial?
5. How does the derivative of $f(x)$ behave at roots with multiplicity greater than 1?

Tip:

Roots with higher multiplicity result in flatter behavior near those points on the graph of the polynomial!