The problem provided seems to be focused on studying the relative position of curves, specifically the function $f(x)$ and its tangent $T$. Let me transcribe and summarize the key tasks from the image for clarity:

Part C: Étude de la position relative de la courbe de la fonction $f$ et de l'une de ses tangentes

1. Montrer que l'équation réduite de la tangente $T$ à la courbe $\mathcal{C}$ en $x = 2$ est : $y = 3x + 3$

2. Montrer qu'étudier la position relative de $\mathcal{C}$ et $T$ revient à étudier le signe de : $\frac{x^3 - 3x^2 + 4}{(x - 2)^2 (x + 1)}$

3. Montrer que, pour tout réel $x$, $x^3 - 3x^2 + 4 = (x - 2)^2 (x + 1)$.

4. En déduire alors la position relative de la courbe $\mathcal{C}$ et de sa tangente $T$ en $x = 2$ sur l'intervalle $[1.5, 5]$.

The problem is asking to study the position of the curve $\mathcal{C}$ defined by the function $f(x) = x^3 - 3x^2 + 4$ and its tangent at $x = 2$, with the following steps:

• Show that the tangent equation at $x = 2$ is $y = 3x + 3$.
• Study the sign of the difference between $f(x)$ and the tangent to understand where the curve is above or below the tangent.
• Prove that the cubic function can be factored as $(x - 2)^2 (x + 1)$.
• Analyze the relative position of the curve and tangent in the interval $[1.5, 5]$.

Would you like me to guide you through these steps in detail or solve one of the specific points?