Question 12 (2 points) Multiply: (4x+9)(-4x^2+x+3) a 21x+27 b -16x^3+21x+27 C -16x^3-32x^2+21x+27 d -16x^3-40x^2+21x+27

To multiply the two polynomials, we use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first polynomial by each term in the second polynomial. <br /><br />$(4x+9)(-4x^{2}+x+3)$<br /><br />First, multiply $4x$ by each term in $(-4x^{2}+x+3)$:<br /><br />$4x \cdot (-4x^{2}) = -16x^{3}$<br /><br />$4x \cdot x = 4x^{2}$<br /><br />$4x \cdot 3 = 12x$<br /><br />Next, multiply $9$ by each term in $(-4x^{2}+x+3)$:<br /><br />$9 \cdot (-4x^{2}) = -36x^{2}$<br /><br />$9 \cdot x = 9x$<br /><br />$9 \cdot 3 = 27$<br /><br />Now, combine like terms:<br /><br />$-16x^{3} + 4x^{2} - 36x^{2} + 12x + 9x + 27$<br /><br />Combine the $x^{2}$ terms and the $x$ terms:<br /><br />$-16x^{3} + (4x^{2} - 36x^{2}) + (12x + 9x) + 27$<br /><br />$-16x^{3} - 32x^{2} + 21x + 27$<br /><br />Therefore, the correct answer is:<br /><br />C $-16x^{3}-32x^{2}+21x+27$