To factor the quadratic equation \(2x^2 + 7x - 15 = 0\), we can look for two numbers that multiply to \(2 \cdot -15 = -30\) and add to \(7\).
The two numbers that work in this case are \(10\) and \(-3\), because:
- \(10 \times -3 = -30\)
- \(10 + (-3) = 7\)
Now, we can rewrite the equation as:
\[2x^2 + 10x - 3x - 15 = 0\]
Next, we can group the terms:
\[ (2x^2 + 10x) + (-3x - 15) = 0 \]
Now, factor out the common terms from each group:
\[ 2x(x + 5) - 3(x + 5) = 0 \]
Now, factor out \(x + 5\):
\[ (2x - 3)(x + 5) = 0 \]
Therefore, the correctly factored form of the equation is \( (2x - 3)(x + 5) = 0 \).
Now, none of the given choices match exactly to this form in the equations provided.
However, if we just rewrite without fully applying the zero product property, we can write it as:
- The roots are indeed related to the factored form and thus \(2x - 3 = 0\) and \(x + 5 = 0\).
So the closest matching factor that fits within is \(2x - 3\).
Thus, the answer from given choices can be interpreted along the format provided but does not directly appear to be \( (2x - 3)(x + 5) = 0 \).
In conclusion, none of the provided options may directly represent the solution accurately due to their format but the closest logical structure led to the factorization presented.
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