Evaluate the zeroes of the polynomial `x^2-3x-m(m+3).`

Options

A. `-m, m-1`
B. `-m, -(mt_3)`
C. `m, mt_3`
D. `-m, m+3`

Answer

To evaluate the zeroes of the polynomial \( x^2 - 3x - m(m + 3) \), we will follow these steps:

### Step 1: Set the polynomial equal to zero
We start by setting the polynomial equal to zero to find the zeroes:
\[ x^2 - 3x - m(m + 3) = 0 \]

### Step 2: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 1 \)
- \( b = -3 \)
- \( c = -m(m + 3) \)

### Step 3: Use the quadratic formula
The zeroes of the polynomial can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot (-m(m + 3))}}{2 \cdot 1} \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ x = \frac{3 \pm \sqrt{9 + 4m(m + 3)}}{2} \]
This simplifies to:
\[ x = \frac{3 \pm \sqrt{9 + 4m^2 + 12m}}{2} \]

### Step 5: Final expression for the zeroes
Thus, the zeroes of the polynomial \( x^2 - 3x - m(m + 3) \) are given by:
\[ x = \frac{3 \pm \sqrt{4m^2 + 12m + 9}}{2} \]

### Summary of the solution
The zeroes of the polynomial \( x^2 - 3x - m(m + 3) \) are:
\[ x = \frac{3 + \sqrt{4m^2 + 12m + 9}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{4m^2 + 12m + 9}}{2} \]