To factor the quadratic expression \(5p^2 + 13p - 6\), we need to find two numbers that multiply to \(5 \times -6 = -30\) and add up to \(13\).
The two numbers that meet these criteria are \(15\) and \(-2\) because:
- \(15 \times -2 = -30\)
- \(15 + (-2) = 13\)
Now we can rewrite the middle term of the quadratic expression using these two numbers:
\[ 5p^2 + 15p - 2p - 6 \]
Next, we can group the terms:
\[ (5p^2 + 15p) + (-2p - 6) \]
Factoring out the common factors in each group gives us:
\[ 5p(p + 3) - 2(p + 3) \]
Now we can factor out the common binomial factor \((p + 3)\):
\[ (5p - 2)(p + 3) \]
Thus, the factored form of \(5p^2 + 13p - 6\) is:
\[ (5p - 2)(p + 3) \]