To solve this problem, we need to determine the quadratic equation in the form $ax^2 + bx + c = 0$ with the given roots $-\frac{3}{4}$ and $5$.

Step 1: Use the relationship between roots and coefficients

For a quadratic equation $ax^2 + bx + c = 0$, the following relationships hold if the roots are $r_1$ and $r_2$:

1. Sum of roots: $r_1 + r_2 = -\frac{b}{a}$
2. Product of roots: $r_1 \cdot r_2 = \frac{c}{a}$

Here, the roots are $r_1 = -\frac{3}{4}$ and $r_2 = 5$.

Compute the sum of the roots:

$r_1 + r_2 = -\frac{3}{4} + 5 = \frac{-3 + 20}{4} = \frac{17}{4}.$

Compute the product of the roots:

$r_1 \cdot r_2 = \left(-\frac{3}{4}\right)(5) = -\frac{15}{4}.$

Step 2: Construct the quadratic equation

Using these values, we can write the quadratic equation as: $a \cdot x^2 + b \cdot x + c = 0 \quad \text{where} \quad b = -a(r_1 + r_2) \quad \text{and} \quad c = a(r_1 \cdot r_2).$

Let $a = 4$ (since all the options use $4x^2$): $b = -4 \cdot \frac{17}{4} = -17, \quad c = 4 \cdot \left(-\frac{15}{4}\right) = -15.$

Thus, the equation becomes: $4x^2 - 17x - 15 = 0.$

Step 3: Match with the given options

The correct answer is: A) $4x^2 - 17x - 15 = 0$.

Would you like a more detailed explanation or have further questions?

Related Questions:

1. How do you verify the roots of a quadratic equation?
2. What is the discriminant of a quadratic equation, and how is it used?
3. How would the equation change if one root were doubled?
4. What happens to the sum and product of roots if the coefficient $a$ changes?
5. How do you factorize a quadratic equation into its root form?

Tip:

For quadratic equations, always double-check the sum and product of roots to ensure correctness!