To factor the quadratic expression \( f^2 + 14f - 15 \), we need to find two numbers that multiply to \(-15\) (the constant term) and add up to \(14\) (the coefficient of the linear term).
The pairs of factors of \(-15\) are:
- \(1\) and \(-15\) (sum: \(-14\))
- \(-1\) and \(15\) (sum: \(14\))
- \(3\) and \(-5\) (sum: \(-2\))
- \(-3\) and \(5\) (sum: \(2\))
The pair that adds up to \(14\) is \(-1\) and \(15\).
Now we can rewrite the expression as:
\[ f^2 + 15f - f - 15 \]
Next, we group the terms:
\[ (f^2 + 15f) + (-f - 15) \]
Now we can factor by grouping:
\[ f(f + 15) - 1(f + 15) \]
This can be factored as:
\[ (f - 1)(f + 15) \]
Thus, the factored form of \( f^2 + 14f - 15 \) is:
\[ (f - 1)(f + 15) \]