Simplify the fractional expression. ((x+h)^3-3(x+h)-(x^3-3x))/(h) square

To simplify the given fractional expression, we need to expand and simplify the then divide by the denominator.<br /><br />Given expression:<br />\[<br />\frac{(x+h)^3 - 3(x+h) - (x^3 - 3x)}{h}<br />\]<br /><br />First, let's expand \((x+h)^3\):<br />\[<br />(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3<br />\]<br /><br />Now substitute this back into the expression:<br />\[<br />\frac{x^3 + 3x^2h + 3xh^2 + h^3 - 3(x+h) - (x^3 - 3x)}{h}<br />\]<br /><br />Simplify the terms inside the numerator:<br />\[<br />= \frac{x^3 + 3x^2h + 3xh^2 + h^3 - 3x - 3h - x^3 + 3x}{h}<br />\]<br /><br />Combine like terms:<br />\[<br />= \frac{3x^2h + 3xh^2 + h^3 - 3h}{h}<br />\]<br /><br />Factor out \(h\) from the numerator:<br />\[<br />= \frac{h(3x^2 + 3xh + h^2 - 3)}{h}<br />\]<br /><br />Cancel the common factor \(h\) in the numerator and denominator:<br />\[<br />= 3x^2 + 3xh + h^2 - 3<br />\]<br /><br />Thus, the simplified expression is:<br />\[<br />\boxed{3x^2 + 3xh + h^2 - 3}<br />\]