Solect the correct choice below and fill in any answer boxes within your choice. A. 12x^2(x+2)-5x(x+2)-3(x+2)=square B. The polynomial is prime.

To solve the expression \( 12x^{2}(x+2) - 5x(x+2) - 3(x+2) \), we can factor by grouping. Notice that each term contains a common factor of \((x+2)\).<br /><br />Let's factor out \((x+2)\) from each term:<br /><br />\[ <br />= (x+2)(12x^2 - 5x - 3)<br />\]<br /><br />Now, we need to check if the quadratic \(12x^2 - 5x - 3\) can be factored further.<br /><br />To factor the quadratic, we look for two numbers that multiply to \(12 \times (-3) = -36\) and add up to \(-5\). The numbers that satisfy this are \(-9\) and \(4\).<br /><br />Rewrite the middle term \(-5x\) using \(-9x\) and \(4x\):<br /><br />\[ <br />12x^2 - 9x + 4x - 3 <br />\]<br /><br />Now, group the terms:<br /><br />\[ <br />= (12x^2 - 9x) + (4x - 3) <br />\]<br /><br />Factor out the greatest common factor from each group:<br /><br />\[ <br />= 3x(4x - 3) + 1(4x - 3) <br />\]<br /><br />Notice that \((4x - 3)\) is a common factor:<br /><br />\[ <br />= (3x + 1)(4x - 3) <br />\]<br /><br />Thus, the original expression can be factored as:<br /><br />\[ <br />(x+2)(3x+1)(4x-3) <br />\]<br /><br />So, the correct choice is:<br /><br />A. \( 12x^{2}(x+2) - 5x(x+2) - 3(x+2) = (x+2)(3x+1)(4x-3) \)