To solve the equation \( x^2 + 17x = -72 \), we first move all terms to one side of the equation to set it to zero:
\[ x^2 + 17x + 72 = 0 \]
Next, we need to factor the quadratic equation \( x^2 + 17x + 72 \). We are looking for two numbers that multiply to \( 72 \) (the constant term) and add up to \( 17 \) (the coefficient of \( x \)).
The pairs of factors of \( 72 \) are:
- \( 1 \times 72 \)
- \( 2 \times 36 \)
- \( 3 \times 24 \)
- \( 4 \times 18 \)
- \( 6 \times 12 \)
- \( 8 \times 9 \)
From the list, we can see that \( 8 \) and \( 9 \) multiply to \( 72 \) and add up to \( 17 \):
\[ 8 + 9 = 17 \]
Now, we can factor the quadratic:
\[ (x + 8)(x + 9) = 0 \]
To find the values of \( x \), we set each factor to zero:
- \( x + 8 = 0 \) leads to \( x = -8 \)
- \( x + 9 = 0 \) leads to \( x = -9 \)
Thus, the solutions to the equation \( x^2 + 17x + 72 = 0 \) are:
\[ \boxed{-8} \quad \text{and} \quad \boxed{-9} \]