Question 5(1 point) Which function is equivalent to f(x)=25x^2+40x+16 a f(x)=(5x+4)(5x-4) b f(x)=(5x-4)^2 C f(x)=(5x+4)^2 d f(x)=(25x+4)(25x-4)

To determine which function is equivalent to \( f(x) = 25x^2 + 40x + 16 \), we need to factor or expand the given options and see which one matches the original function.<br /><br />Let's start by expanding each option:<br /><br />**Option a: \( f(x) = (5x+4)(5x-4) \)**<br /><br />Using the difference of squares formula:<br />\[ (a+b)(a-b) = a^2 - b^2 \]<br /><br />Here, \( a = 5x \) and \( b = 4 \):<br />\[ (5x+4)(5x-4) = (5x)^2 - 4^2 = 25x^2 - 16 \]<br /><br />This does not match \( f(x) = 25x^2 + 40x + 16 \).<br /><br />**Option b: \( f(x) = (5x-4)^2 \)**<br /><br />Expanding this:<br />\[ (5x-4)^2 = (5x-4)(5x-4) \]<br />\[ = (5x)^2 - 2(5x)(4) + 4^2 \]<br />\[ = 25x^2 - 40x + 16 \]<br /><br />This matches \( f(x) = 25x^2 + 40x + 16 \).<br /><br />**Option c: \( f(x) = (5x+4)^2 \)**<br /><br />Expanding this:<br />\[ (5x+4)^2 = (5x+4)(5x+4) \]<br />\[ = (5x)^2 + 2(5x)(4) + 4^2 \]<br />\[ = 25x^2 + 40x + 16 \]<br /><br />This also matches \( f(x) = 25x^2 + 40x + 16 \).<br /><br />**Option d: \( f(x) = (25x+4)(25x-4) \)**<br /><br />Using the difference of squares formula:<br />\[ (a+b)(a-b) = a^2 - b^2 \]<br /><br />Here, \( a = 25x \) and \( b = 4 \):<br />\[ (25x+4)(25x-4) = (25x)^2 - 4^2 = 625x^2 - 16 \]<br /><br />This does not match \( f(x) = 25x^2 + 40x + 16 \).<br /><br />Therefore, the correct answers are:<br />b) \( f(x) = (5x-4)^2 \)<br />c) \( f(x) = (5x+4)^2 \)<br /><br />Both options b and c are equivalent to \( f(x) = 25x^2 + 40x + 16 \).