Question 1209534
Here's how to find the range of values for k:

**1. Rewrite the expression:**

The given expression is 3 - 4k - (k+3)x - x². We can rewrite this as:

-x² - (k+3)x + (3 - 4k)

**2. Consider the quadratic form:**

This is a quadratic expression in x of the form ax² + bx + c, where:

* a = -1
* b = -(k+3)
* c = (3 - 4k)

**3. Condition for negativity:**

For the quadratic expression to be negative for all real values of x, two conditions must be met:

* **The parabola must open downwards:** This is already satisfied since a = -1 (negative).
* **The quadratic must have no real roots:**  This means the discriminant (b² - 4ac) must be less than zero.

**4. Apply the discriminant condition:**

The discriminant is:

b² - 4ac = [-(k+3)]² - 4*(-1)*(3 - 4k)
        = (k² + 6k + 9) + 4(3 - 4k)
        = k² + 6k + 9 + 12 - 16k
        = k² - 10k + 21

For no real roots, the discriminant must be less than zero:

k² - 10k + 21 < 0

**5. Solve the inequality:**

Factor the quadratic:

(k - 3)(k - 7) < 0

This inequality is satisfied when k is between the roots 3 and 7.

**6. Final answer:**

Therefore, the expression 3 - 4k - (k+3)x - x² will be negative for all real values of x when:

3 < k < 7