What should be added to 3x + 7 to make it equal to x^2 + 4x + 1?

To determine what needs to be added to the expression 3x + 7 to equal the expression x² + 4x + 1, we can set up the equation:

3x + 7 + ? = x² + 4x + 1

To isolate the unknown (let’s call it k), we rearrange the equation:

k = (x² + 4x + 1) – (3x + 7)

Now, let’s simplify the right side:

• Distributing the negative sign: k = x² + 4x + 1 – 3x – 7
• Combine like terms: k = x² + (4x – 3x) + (1 – 7)
• This simplifies to: k = x² + x – 6

Therefore, the expression that must be added to 3x + 7 in order to yield x² + 4x + 1 is x² + x – 6.