(3) \( 12 x^{3}-4 x \) (7) \( 3 a^{2}-9 a b+12 a^{2} b \) \( -2 m n+28 m^{2} n-6 n \)

Let's factor each of the given expressions by identifying and extracting their greatest common factors (GCF) and then factoring further where possible. --- ### (3) \( 12x^{3} - 4x \) **Step 1:** Identify the GCF. - The GCF of \(12x^3\) and \(4x\) is \(4x\). **Step 2:** Factor out the GCF. \[ 12x^{3} - 4x = 4x(3x^{2} - 1) \] **Factored Form:** \[ 12x^{3} - 4x = 4x(3x^{2} - 1) \] --- ### (7) \( 3a^{2} - 9ab + 12a^{2}b \) **Step 1:** Identify the GCF. - All terms have a common factor of \(3a\). **Step 2:** Factor out the GCF. \[ 3a^{2} - 9ab + 12a^{2}b = 3a(a - 3b + 4a b) \] **Alternative Factoring:** You can also rearrange and factor further if desired: \[ 3a(a - 3b + 4a b) = 3a(a(1 + 4b) - 3b) \] **Factored Form:** \[ 3a^{2} - 9ab + 12a^{2}b = 3a(a - 3b + 4ab) \] --- ### \( -2mn + 28m^{2}n - 6n \) **Step 1:** Identify the GCF. - All terms have a common factor of \(2n\). **Step 2:** Factor out the GCF. \[ -2mn + 28m^{2}n - 6n = 2n(-m + 14m^{2} - 3) \] **Step 3:** Factor the quadratic expression inside the parentheses. \[ 14m^{2} - m - 3 \] Find two numbers that multiply to \(-42\) (14 * -3) and add to \(-1\). These numbers are \(-7\) and \(6\). \[ 14m^{2} - m - 3 = 14m^{2} - 7m + 6m - 3 = 7m(2m - 1) + 3(2m - 1) = (7m + 3)(2m - 1) \] **Final Factored Form:** \[ -2mn + 28m^{2}n - 6n = 2n(7m + 3)(2m - 1) \] --- **Summary of Factored Forms:** 1. \( 12x^{3} - 4x = 4x(3x^{2} - 1) \) 2. \( 3a^{2} - 9ab + 12a^{2}b = 3a(a - 3b + 4ab) \) 3. \( -2mn + 28m^{2}n - 6n = 2n(7m + 3)(2m - 1) \)