Factor. \( 2 a^{2}-3 a-20 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( 2 a^{2}-3 a-20=\square \) B. The trinomial is not factorable.

To factor the trinomial \( 2a^{2}-3a-20 \), we can look for two numbers that multiply to \( 2 \times -20 = -40 \) and add to \( -3 \). These two numbers are \( 5 \) and \( -8 \). We can then rewrite the middle term and factor by grouping: \( 2a^{2} + 5a - 8a - 20 \) \( = a(2a + 5) - 4(2a + 5) \) \( = (2a + 5)(a - 4) \) So, we have: \( 2 a^{2}-3 a-20=(2a + 5)(a - 4) \) The correct option is A: \( 2 a^{2}-3 a-20=(2a + 5)(a - 4) \). Factoring is a great skill in mathematics, allowing you to simplify expressions and solve equations more easily. When you can factor a quadratic expression, you can quickly find its roots using the zero-product property, which states that if the product of two factors equals zero, then at least one of the factors must be zero. Many people make the mistake of assuming all quadratic equations can be factored neatly. However, some quadratics may involve irrational or complex numbers as roots, making them not easily factorizable over the integers. Always check the discriminant \( b^2 - 4ac \) to see if it's a perfect square before proceeding to factor!