Find a quadratic polynomial, the sum of whose zeroes is `-(3)/(4)` and product is 5.

Explanation

To find a quadratic polynomial whose sum of zeroes is \(-\frac{3}{4}\) and product is \(5\), we can follow these steps:

### Step 1: Understand the general form of a quadratic polynomial
A quadratic polynomial can be expressed in the form:
\[ P(x) = x^2 - (\text{sum of zeroes}) \cdot x + (\text{product of zeroes}) \]
Let the zeroes of the polynomial be \(\alpha\) and \(\beta\).

### Step 2: Substitute the given values
From the question, we know:
- Sum of zeroes (\(\alpha + \beta\)) = \(-\frac{3}{4}\)
- Product of zeroes (\(\alpha \beta\)) = \(5\)

Substituting these values into the general form of the polynomial gives:
\[ P(x) = x^2 - \left(-\frac{3}{4}\right) x + 5 \]

### Step 3: Simplify the polynomial
This simplifies to:
\[ P(x) = x^2 + \frac{3}{4} x + 5 \]

### Step 4: Eliminate the fraction
To eliminate the fraction, we can multiply the entire polynomial by \(4\) (the denominator of the fraction):
\[ 4P(x) = 4\left(x^2 + \frac{3}{4} x + 5\right) \]
This results in:
\[ 4P(x) = 4x^2 + 3x + 20 \]

### Step 5: Write the final polynomial
Thus, the quadratic polynomial we are looking for is:
\[ P(x) = 4x^2 + 3x + 20 \]

### Summary
The quadratic polynomial whose sum of zeroes is \(-\frac{3}{4}\) and product is \(5\) is:
\[ P(x) = 4x^2 + 3x + 20 \]

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