12. Find and simplify the difference quotient for f(x)=2x^2+3x-6 a. 4x+2h+3 b. 2x+2h+3 c. 4xh+2h^2+3h d. 2h+3

To find and simplify the difference quotient for \( f(x) = 2x^2 + 3x - 6 \), we need to compute the expression:<br /><br />\[<br />\frac{f(x+h) - f(x)}{h}<br />\]<br /><br />First, let's find \( f(x+h) \):<br /><br />\[<br />f(x+h) = 2(x+h)^2 + 3(x+h) - 6<br />\]<br /><br />Expanding \( (x+h)^2 \):<br /><br />\[<br />(x+h)^2 = x^2 + 2xh + h^2<br />\]<br /><br />So,<br /><br />\[<br />f(x+h) = 2(x^2 + 2xh + h^2) + 3(x+h) - 6<br />\]<br />\[<br />= 2x^2 + 4xh + 2h^2 + 3x + 3h - 6<br />\]<br /><br />Next, we subtract \( f(x) \) from \( f(x+h) \):<br /><br />\[<br />f(x+h) - f(x) = (2x^2 + 4xh + 2h^2 + 3x + 3h - 6) - (2x^2 + 3x - 6)<br />\]<br /><br />Simplify by canceling out the common terms:<br /><br />\[<br />f(x+h) - f(x) = 2x^2 + 4xh + 2h^2 + 3x + 3h - 6 - 2x^2 - 3x + 6<br />\]<br />\[<br />= 4xh + 2h^2 + 3h<br />\]<br /><br />Now, divide by \( h \):<br /><br />\[<br />\frac{f(x+h) - f(x)}{h} = \frac{4xh + 2h^2 + 3h}{h}<br />\]<br /><br />Simplify by factoring out \( h \) in the numerator:<br /><br />\[<br />= \frac{h(4x + 2h + 3)}{h}<br />\]<br /><br />Cancel \( h \):<br /><br />\[<br />= 4x + 2h + 3<br />\]<br /><br />Thus, the simplified difference quotient is:<br /><br />\[<br />4x + 2h + 3<br />\]<br /><br />The correct answer is:<br /><br />\[<br />a. \quad 4x + 2h + 3<br />\]