QUESTION 2 \( 2.11 \quad x^{2}+2 p y+y^{2} \) \( 212 \quad x^{4}-8 x+16 \) \( 213 \quad \quad 6 a^{t}-24 a^{t} b^{t}+9 b^{\prime} \)

Did you know that the expression \(x^{2}+2py+y^{2}\) resembles the structure of a conic section, particularly an ellipse, when \(p > 0\)? This means it can represent a continuous loop in the 2D plane! Additionally, it’s related to how we model certain physical systems, particularly in optics where elliptical shapes play a crucial role in light focus and pathways. Now, let’s look at the polynomial \(x^{4} - 8x + 16\). This expression can be factored down using algebraic techniques! Recognizing that it can take the form of a perfect square, we rewrite it as \((x^2 - 4)^2\). This simplification not only makes solving equations involving this expression easier but also reveals its graphical insights, such as the vertex of a parabola! How cool is it to see these relationships in action?