1)) Divide . Write your answer in simplest form. (3x^3+x^2)/(12x^2)+11x+2div (2)/(16x^2)-1 square

To divide the given expression, we need to multiply the first fraction by the reciprocal of the second fraction. Let's start by finding the reciprocal of the second fraction.<br /><br />The reciprocal of $\frac{2}{16x^2 - 1}$ is $\frac{16x^2 - 1}{2}$.<br /><br />Now, we can rewrite the original expression as:<br /><br />$\frac{3x^3 + x^2}{12x^2 + 11x + 2} \times \frac{16x^2 - 1}{2}$<br /><br />Next, we can simplify the expression by factoring the numerator and denominator where possible.<br /><br />The numerator $3x^3 + x^2$ can be factored as $x^2(3x + 1)$.<br /><br />The denominator $12x^2 + 11x + 2$ can be factored as $(4x + 1)(3x + 2)$.<br /><br />The denominator $16x^2 - 1$ can be factored as $(4x + 1)(4x - 1)$.<br /><br />Now, we can rewrite the expression as:<br /><br />$\frac{x^2(3x + 1)}{(4x + 1)(3x + 2)} \times \frac{(4x + 1)(4x - 1)}{2}$<br /><br />We can cancel out the common factors in the numerator and denominator:<br /><br />$\frac{x^2(3x + 1)}{(4x + 1)(3x + 2)} \times \frac{(4x + 1)(4x - 1)}{2} = \frac{x^2(3x + 1)(4x - 1)}{2(3x + 2)}$<br /><br />Finally, we can simplify the expression further by canceling out the common factor of $3x + 1$:<br /><br />$\frac{x^2(3x + 1)(4x - 1)}{2(3x + 2)} = \frac{x^2(4x - 1)}{2(3x + 2)}$<br /><br />Therefore, the simplified form of the given expression is $\frac{x^2(4x - 1)}{2(3x + 2)}$.