Factor the trinomial \[ 2 x^{2} z+20 x z+42 z \] Seleci the correct choice below and, if necessary, fil in the answer box to complete your choice. A. \( 2 x^{2} z+20 x z+42 z= \) (Factor completey) B. \( 2 x^{2} z+20 x z+42 z \) is prime Helo mo solun this

To factor the trinomial \( 2x^{2}z + 20xz + 42z \), we can start by looking for a common factor in all the terms. Notice that \( z \) can be factored out: \[ 2x^{2}z + 20xz + 42z = z(2x^{2} + 20x + 42) \] Next, we can focus on factoring the quadratic \( 2x^{2} + 20x + 42 \). First, we can find the greatest common factor (GCF) of the coefficients 2, 20, and 42, which is 2: \[ 2(x^{2} + 10x + 21) \] Now we need to factor the quadratic expression \( x^{2} + 10x + 21 \). We can look for two numbers that multiply to 21 and add up to 10. Those numbers are 3 and 7: \[ x^{2} + 10x + 21 = (x + 3)(x + 7) \] Putting it all together, we have: \[ 2(x + 3)(x + 7) \] So, the complete factorization of the original trinomial is: \[ 2x^{2}z + 20xz + 42z = z \cdot 2(x + 3)(x + 7) \] Thus, the answer for part A is: \( z \cdot 2(x + 3)(x + 7) \)