To determine which quadratic equation has the roots -4 and -5, we'll follow the standard process for forming a quadratic equation given its roots.
Given roots [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex]:
[tex]\[ r_1 = -4 \][/tex]
[tex]\[ r_2 = -5 \][/tex]
The quadratic equation having roots [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex] is given by the formula:
[tex]\[ x^2 - (r_1 + r_2)x + r_1r_2 = 0 \][/tex]
### Step 1: Calculate the sum of the roots.
[tex]\[ r_1 + r_2 = -4 + (-5) = -9 \][/tex]
### Step 2: Calculate the product of the roots.
[tex]\[ r_1 \cdot r_2 = (-4) \cdot (-5) = 20 \][/tex]
### Step 3: Form the quadratic equation.
Using the values found:
[tex]\[ x^2 - (r_1 + r_2)x + r_1r_2 = 0 \][/tex]
Substituting the sum and product:
[tex]\[ x^2 - (-9)x + 20 = 0 \][/tex]
[tex]\[ x^2 + 9x + 20 = 0 \][/tex]
Thus, the quadratic equation with roots -4 and -5 is:
[tex]\[ x^2 + 9x + 20 = 0 \][/tex]
Comparing this equation with the given choices, the correct equation is:
[tex]\[ \boxed{x^2 + 9x + 20 = 0} \][/tex]
So, the correct choice is the fourth option:
[tex]\[ x^2 + 9x + 20 = 0 \][/tex]
