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The problem presented in the image involves finding values of \(a\) and \(b\) such that \(g(x) = bxf(x)\) is a primitive (or antiderivative) of \(f(x) = x^2 + ax\). Here are the steps to solve this problem:

1. **Express \(g(x)\) in terms of \(f(x)\):**
   \[
   g(x) = bxf(x) = bx(x^2 + ax) = bx^3 + bax^2
   \]

2. **Find the derivative of \(g(x)\):**
   Using the power rule, \(g'(x) = 3bx^2 + 2bax\).

3. **Set \(g'(x)\) equal to \(f(x)\) and compare coefficients:**
   \[
   g'(x) = 3bx^2 + 2bax \quad \text{and} \quad f(x) = x^2 + ax
   \]
   Setting these equal to each other, we get:
   \[
   3bx^2 + 2bax = x^2 + ax
   \]

   Equating coefficients, we find:
   - For \(x^2\): \(3b = 1\) \(\Rightarrow b = \frac{1}{3}\)
   - For \(x\): \(2ba = a\) \(\Rightarrow 2b = 1\) (as \(a \neq 0\) to avoid trivial solution)

   Given \(b = \frac{1}{3}\), \(2 \times \frac{1}{3} = \frac{2}{3}\) which does not satisfy \(2b = 1\). This inconsistency suggests there might be a mistake in interpreting the roles of \(a\) and \(b\).

   Let's reevaluate:
   - Setting \(b = \frac{1}{3}\) makes \(3b = 1\), which is consistent for the \(x^2\) coefficient.
   - The coefficient of \(x\) leads to an issue since \(2ba\) should equal \(a\). This only holds if \(a = 0\) (or the initial problem setup might have an error or additional condition).

Based on this analysis:
- \(b = \frac{1}{3}\)
- \(a = 0\) to satisfy the equation with the coefficients of \(x\).

Would you like to explore more about this solution or have any further questions on this topic? Here are some related questions you might consider:
1. How would the solution change if \(a\) and \(b\) had different constraints?
2. What would be the impact of adding another term to \(f(x)\), say \(cx^3\)?
3. Can you explain the concept of a primitive function in more detail?
4. How can the power rule be used to differentiate polynomials of higher degrees?
5. What are some common methods for integrating polynomial functions?

Tip: Remember to always check for consistency across all terms in an equation when equating coefficients, especially in contexts involving derivatives and antiderivatives.

Fie f, g : ℝ → ℝ, f(x) = x² + ax, g(x) = bxf(x). Să se determine a, b ∈ ℝ pentru care funcția g este o primitivă a funcției f.

Solve the problem where g(x) = bxf(x) is a primitive of f(x) = x² + ax by determining values for a and b. This involves using differentiation, integration, and polynomial functions.

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