The image shows a section of a lab assignment, which includes instructions for working with polynomials $P(x) = 2x^3 + x^2 + 3x + 4$ and $Q(x) = x^3 - 5x + 3$. The tasks involve operations like adding, multiplying, and graphing these polynomials in various ways. Here's a breakdown of the tasks:

1. Addition of Polynomials: Add $P(x)$ and $Q(x)$, graph them individually and their sum on the same set of axes in different colors.

2. Adding Zero Polynomial: Let $0(x) = 0x^3 + 0x^2 + 0x + 0$. Graph $P(x)$, $Q(x)$, and $0(x)$ to explore the effect of adding zero to a polynomial.

3. Finding R(x): Find a polynomial $R(x)$ such that $P + R = 0$, and graph all three polynomials on the same set of axes, using different line characteristics.

4. Multiplying by a Scalar: Multiply the polynomial $P(x)$ by 5 (i.e., $5P(x)$) and graph a series of scaled versions ($-5P, -4P, \dots, 5P$) using a loop to generate the graphics.

5. Combining with Real Numbers: Choose three linear combinations of $P$ and $Q$ with real numbers $a$ and $b$, neither being zero, and graph them.

6. Spanning a Space: Show that $P$ and $Q$ span a two-dimensional space by identifying a matrix with columns corresponding to $P(x)$ and $Q(x)$.

7. Existence of Higher Degree Polynomial: Investigate whether a degree 2 polynomial exists in the span of $P(x)$ and $Q(x)$.

Would you like help solving any of these specific tasks or need detailed explanations for any particular section? Let me know!

Here are five related questions to extend your understanding:

1. How do you perform polynomial addition algebraically?
2. What is the geometric interpretation of adding two polynomials?
3. How does scaling a polynomial by a real number affect its graph?
4. What is the significance of finding a polynomial $R(x)$ such that $P + R = 0$?
5. Can you graph multiple functions programmatically using MATLAB or Python?

Tip: When adding polynomials, remember that only like terms (same degree) are combined.