Question: Factor the trinomial. $54x^2 + 30x + 4$ Select the correct choice below and, if necessary, fill…

Factor the trinomial.

$54x^2 + 30x + 4$

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. $54x^2 + 30x + 4 = \boxed{\phantom{0}}$ (Factor completely.)

B. The trinomial is prime.

Solution

To factor the trinomial \(54x^2 + 30x + 4\) completely, follow these steps: First, identify the coefficients: \[ a = 54, \quad b = 30, \quad c = 4 \] Next, calculate the product of \(a\) and \(c\): \[ a \times c = 54 \times 4 = 216 \] Find two numbers that multiply to \(216\) and add up to \(30\). The numbers are \(12\) and \(18\): \[ 12 \times 18 = 216 \quad \text{and} \quad 12 + 18 = 30 \] Split the middle term using these numbers: \[ 54x^2 + 12x + 18x + 4 \] Group the terms: \[ (54x^2 + 12x) + (18x + 4) \] Factor out the greatest common factor from each group: \[ 6x(9x + 2) + 2(9x + 2) \] Factor out the common binomial factor \((9x + 2)\): \[ (6x + 2)(9x + 2) \] Factor out the greatest common factor from the first binomial: \[ 2(3x + 1)(9x + 2) \] Final Factored Form: \[ 2(3x + 1)(9x + 2) \]