5.54. Factor the following: (a) \( 2 x^{2}+3 x-2 \)

To factor the quadratic \( 2x^{2} + 3x - 2 \), we look for two numbers that multiply to \( 2 \times -2 = -4 \) and add up to \( 3 \). The numbers \( 4 \) and \( -1 \) fit perfectly! We can rewrite the middle term: \[ 2x^2 + 4x - x - 2 \] Now, group the terms: \[ (2x^2 + 4x) + (-x - 2) \] Factor out common terms: \[ 2x(x + 2) - 1(x + 2) \] Now, we can see a common factor of \( (x + 2) \): \[ (2x - 1)(x + 2) \] So, \( 2x^{2} + 3x - 2 = (2x - 1)(x + 2) \). To verify, you can easily expand it back, and you'll get your original expression! Happy factoring!