Check whether the following is a quadratic equation:
`(x-3)(2x+1)=x(x+5)`

Answer

To determine whether the equation \((x-3)(2x+1) = x(x+5)\) is a quadratic equation, we will first expand both sides and then rearrange the equation into standard form.

### Step 1: Expand both sides of the equation

Starting with the left-hand side:
\[ (x-3)(2x+1) = x \cdot 2x + x \cdot 1 - 3 \cdot 2x - 3 \cdot 1 \]
\[ = 2x^2 + x - 6x - 3 \]
\[ = 2x^2 - 5x - 3 \]

Now, expanding the right-hand side:
\[ x(x+5) = x^2 + 5x \]

### Step 2: Set the equation to zero

Now we have:
\[ 2x^2 - 5x - 3 = x^2 + 5x \]
To set the equation to zero, we will move all terms to one side:
\[ 2x^2 - 5x - 3 - x^2 - 5x = 0 \]
This simplifies to:
\[ (2x^2 - x^2) + (-5x - 5x) - 3 = 0 \]
\[ x^2 - 10x - 3 = 0 \]

### Step 3: Identify the degree of the polynomial

The resulting equation is:
\[ x^2 - 10x - 3 = 0 \]
The highest power of \(x\) in this equation is 2.

### Conclusion

Since the highest power of \(x\) is 2, the equation \(x^2 - 10x - 3 = 0\) is indeed a quadratic equation.