Divide. Express the quotient in simplest form. (x^2+x-12)/(2x)div (x^2-5x+6)/(6x^2) (x-2)/(3x^2)+12x (3x(x^2+x-12))/(x^2)-5x+6 (3x+12)/(-2) (3x^2+12x)/(x-2)

To solve the problem, we need to divide the given expressions and simplify the quotient.<br /><br />Given:<br />\[ \frac{x^{2}+x-12}{2 x} \div \frac{x^{2}-5 x+6}{6 x^{2}} \]<br /><br />First, rewrite the division as multiplication by the reciprocal:<br />\[ \frac{x^{2}+x-12}{2 x} \times \frac{6 x^{2}}{x^{2}-5 x+6} \]<br /><br />Next, factorize the numerators and denominators where possible:<br />\[ x^{2} + x - 12 = (x + 4)(x - 3) \]<br />\[ x^{2} - 5 x + 6 = (x - 2)(x - 3) \]<br /><br />Substitute these factorizations back into the expression:<br />\[ \frac{(x + 4)(x - 3)}{2 x} \times \frac{6 x^{2}}{(x - 2)(x - 3)} \]<br /><br />Cancel out common factors in the numerator and denominator:<br />\[ \frac{(x + 4)(x - 3)}{2 x} \times \frac{6 x^{2}}{(x - 2)(x - 3)} = \frac{(x + 4)}{2 x} \times \frac{6 x}{(x - 2)} \]<br /><br />Simplify the remaining expression:<br />\[ \frac{(x + 4)}{2 x} \times \frac{6 x}{(x - 2)} = \frac{(x + 4) \cdot 6 x}{2 x \cdot (x - 2)} \]<br /><br />Cancel out \( x \) from the numerator and denominator:<br />\[ \frac{6(x + 4)}{2(x - 2)} \]<br /><br />Simplify the fraction:<br />\[ \frac{6(x + 4)}{2(x - 2)} = \frac{3(x + 4)}{x - 2} \]<br /><br />Thus, the quotient in simplest form is:<br />\[ \frac{3(x + 4)}{x - 2} \]