Find a quadratic polynomial, the sum and product of whose zeroes are $-3$ and $2$, respectively.

Step 1: Define the quadratic polynomial and its zeroes

We start by defining a general quadratic polynomial as $P(x) = ax^2 + bx + c$. We also denote its two zeroes as $\alpha$ and $\beta$. This is a standard way to represent a quadratic polynomial and its roots.

Step 2: State the given sum and product of zeroes

The problem provides us with the sum of the zeroes as $-3$ and the product of the zeroes as $2$. We write these relationships mathematically using our defined zeroes $\alpha$ and $\beta$.

Step 3: Relate zeroes to coefficients

We recall the general relationships between the zeroes and coefficients of a quadratic polynomial. The sum of the zeroes is equal to $-b/a$, and the product of the zeroes is equal to $c/a$.

Step 4: Equate the given values with the coefficient relationships

Now we equate the given sum and product values with their respective coefficient relationships. This allows us to establish connections between the coefficients $a, b, c$ and the given information.

Step 5: Choose a value for 'a'

To find a specific quadratic polynomial, we can choose a value for $a$. The simplest choice is $a=1$, which makes the calculations straightforward. Any non-zero value of $a$ would result in a valid polynomial, but $a=1$ gives the simplest form.

Step 6: Calculate 'b' and 'c'

Using $a=1$, we substitute this value into the equations derived in the previous step. This allows us to directly calculate the values of $b$ and $c$ which are $3$ and $2$ respectively.

Step 7: Form the quadratic polynomial

Finally, we substitute the determined values of $a=1$, $b=3$, and $c=2$ back into the general form of the quadratic polynomial $ax^2 + bx + c$. This yields the desired quadratic polynomial.

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