Factor the trinomial. \[ 7 x^{2}+33 x-10 \]

To factor the trinomial \( 7x^{2} + 33x - 10 \), we start by multiplying the coefficient of \( x^{2} \) (which is 7) by the constant term (which is -10). This gives us \( 7 \times -10 = -70 \). Next, we need two numbers that multiply to -70 and add up to 33. The numbers 35 and -2 meet those criteria since \( 35 \times -2 = -70 \) and \( 35 + (-2) = 33 \). Now we can rewrite the middle term (33x) using these two numbers: \[ 7x^{2} + 35x - 2x - 10 \] Next, we group the terms: \[ (7x^{2} + 35x) + (-2x - 10) \] Factoring out the common factors in each group gives us: \[ 7x(x + 5) - 2(x + 5) \] Now, we can factor out the common binomial \( (x + 5) \): \[ (7x - 2)(x + 5) \] Thus, the factored form of the trinomial \( 7x^{2} + 33x - 10 \) is: \[ (7x - 2)(x + 5) \]