Question 9/Multiple Choice Worth 1 points) (07.03 MC) Factor completely 2x^2+9x+4 (2x+2)(x+2) (2x+1)(x+4) (2x+4)(x+1) (2x+2)(x+4)

To factor the quadratic expression \(2x^2 + 9x + 4\) completely, we need to find two binomials whose product gives us the original quadratic expression.<br /><br />First, let's rewrite the quadratic expression in standard form:<br />\[ 2x^2 + 9x + 4 \]<br /><br />We need to find two numbers that multiply to \(2 \cdot 4 = 8\) (the product of the coefficient of \(x^2\) and the constant term) and add up to \(9\) (the coefficient of \(x\)).<br /><br />The numbers that satisfy these conditions are \(1\) and \(8\), because:<br />\[ 1 \cdot 8 = 8 \]<br />\[ 1 + 8 = 9 \]<br /><br />Next, we split the middle term \(9x\) using these two numbers:<br />\[ 2x^2 + 9x + 4 = 2x^2 + 8x + x + 4 \]<br /><br />Now, we group the terms in pairs:<br />\[ (2x^2 + 8x) + (x + 4) \]<br /><br />Factor out the greatest common factor (GCF) from each pair:<br />\[ 2x(x + 4) + 1(x + 4) \]<br /><br />Notice that \((x + 4)\) is a common factor:<br />\[ (2x + 1)(x + 4) \]<br /><br />So, the completely factored form of \(2x^2 + 9x + 4\) is:<br />\[ (2x + 1)(x + 4) \]<br /><br />Therefore, the correct answer is:<br />\[ \boxed{(2x + 1)(x + 4)} \]