and LCM of the polynomials \( 6\left(x^{2}+6 x+8\right) \) and \( 2 x^{2}-32 \)

To find the LCM of the polynomials \( 6\left(x^{2}+6 x+8\right) \) and \( 2 x^{2}-32 \), we first need to factor each polynomial. Starting with \( 6\left(x^{2}+6 x+8\right) \): 1. Factor \( x^2 + 6x + 8 \). - This can be factored as \( (x+2)(x+4) \). 2. So, \( 6(x^2 + 6x + 8) = 6(x + 2)(x + 4) \). Now for \( 2x^{2}-32 \): 1. Factor out the common factor \( 2 \): - This gives \( 2(x^2 - 16) \). 2. \( x^2 - 16 \) is a difference of squares: - Factored, it becomes \( (x - 4)(x + 4) \). 3. Therefore, \( 2x^{2} - 32 = 2(x - 4)(x + 4) \). Now we have: - \( 6(x + 2)(x + 4) \) - \( 2(x - 4)(x + 4) \) To find the LCM, we need to take each factor at its highest power: - From \( 6(x + 2)(x + 4) \), we take \( 6 \) and \( (x + 2) \). - From \( 2(x - 4)(x + 4) \), we take \( 2 \) and \( (x - 4) \), and since \( (x + 4) \) is shared between both, we just take it once. Now, check the maximum of coefficients: - The maximum of \( 6 \) and \( 2 \) is \( 6 \). Putting it all together: \[ \text{LCM} = 6 \cdot (x + 2) \cdot (x - 4) \cdot (x + 4) \] This simplifies to: \[ \text{LCM} = 6(x + 2)(x - 4)(x + 4) \] Thus, the least common multiple (LCM) of the given polynomials is: \[ 6(x + 2)(x - 4)(x + 4) \]