[tex]f(x) = 3 {x}^{2} + 5x + 1 \\ \\ \frac{f(x + h) - f(x)}{h} \\ \\ \frac{(3(x + h)^{2} + 5(x + h) + 1) - (3 {x}^{2} + 5x + 1 }{h} \\ \\ \frac{(3( {x}^{2} + 2xh + {h}^{2}) + 5x + 5h + 1) - 3 {x}^{2} - 5x - 1}{h} \\ \\ \frac{3 {x}^{2} + 6xh + 3 {h}^{2} + 5x + 5h + 1 - 3 {x}^{2} - 5x - 1 }{h} \\ \\ \frac{6xh + 3 {h}^{2} + 5h}{h} \\ \\ \frac{h(6x + 3h + 5)}{h} \\ \\ = 6x + 3h + 5[/tex]
