Which expression is equivalent to 20x^2+17x+3 ? a (4x+3)(5x-1) b (5x-3)(4x-1) c (4x-3)(5x+1) d (5x+3)(4x+1)

To determine which expression is equivalent to \(20x^2 + 17x + 3\), we need to factor the quadratic expression. We will test each option by expanding it and checking if it matches the given quadratic expression.<br /><br />Let's start with option (a):<br />\[<br />(4x + 3)(5x - 1)<br />\]<br /><br />Expanding this:<br />\[<br />(4x + 3)(5x - 1) = 4x \cdot 5x + 4x \cdot (-1) + 3 \cdot 5x + 3 \cdot (-1)<br />\]<br />\[<br />= 20x^2 - 4x + 15x - 3<br />\]<br />\[<br />= 20x^2 + 11x - 3<br />\]<br /><br />This does not match \(20x^2 + 17x + 3\).<br /><br />Next, let's try option (b):<br />\[<br />(5x - 3)(4x - 1)<br />\]<br /><br />Expanding this:<br />\[<br />(5x - 3)(4x - 1) = 5x \cdot 4x + 5x \cdot (-1) + (-3) \cdot 4x + (-3) \cdot (-1)<br />\]<br />\[<br />= 20x^2 - 5x - 12x + 3<br />\]<br />\[<br />= 20x^2 - 17x + 3<br />\]<br /><br />This does not match \(20x^2 + 17x + 3\).<br /><br />Next, let's try option (c):<br />\[<br />(4x - 3)(5x + 1)<br />\]<br /><br />Expanding this:<br />\[<br />(4x - 3)(5x + 1) = 4x \cdot 5x + 4x \cdot 1 + (-3) \cdot 5x + (-3) \cdot 1<br />\]<br />\[<br />= 20x^2 + 4x - 15x - 3<br />\]<br />\[<br />= 20x^2 - 11x - 3<br />\]<br /><br />This does not match \(20x^2 + 17x + 3\).<br /><br />Finally, let's try option (d):<br />\[<br />(5x + 3)(4x + 1)<br />\]<br /><br />Expanding this:<br />\[<br />(5x + 3)(4x + 1) = 5x \cdot 4x + 5x \cdot 1 + 3 \cdot 4x + 3 \cdot 1<br />\]<br />\[<br />= 20x^2 + 5x + 12x + 3<br />\]<br />\[<br />= 20x^2 + 17x + 3<br />\]<br /><br />This matches \(20x^2 + 17x + 3\).<br /><br />Therefore, the correct answer is:<br />\[<br />\boxed{(5x + 3)(4x + 1)}<br />\]