To work with the expression \( x^2 + 15x + 56 \), we can factor it if possible.
We are looking for two numbers that multiply together to give \( 56 \) (the constant term) and add up to \( 15 \) (the coefficient of the linear term).
The numbers \( 7 \) and \( 8 \) satisfy these conditions:
- \( 7 \times 8 = 56 \)
- \( 7 + 8 = 15 \)
Thus, we can factor the quadratic expression as follows:
\[ x^2 + 15x + 56 = (x + 7)(x + 8) \]
If you'd like to solve for the roots of the equation \( x^2 + 15x + 56 = 0 \), you can set the factored form equal to zero:
\[ (x + 7)(x + 8) = 0 \]
This gives the solutions:
\[ x + 7 = 0 \quad \Rightarrow \quad x = -7 \] \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \]
So the roots of the quadratic equation are \( x = -7 \) and \( x = -8 \).