Solve f(x+h)-f(x)/h=[2(x+h)^2-5(x+1)+1]-(2x^2-5x+1)

$$\frac { f ( x + h ) - f ( x ) } { h } = [ 2 ( x + h ) ^ { 2 } - 5 ( x + 1 ) + 1 ] - ( 2 x ^ { 2 } - 5 x + 1 )$$

Solve for f

$$f=4hx+2h^{2}-5$$

$$h\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by h.

$$f\left(x+h\right)-fx=h\left(2\left(x+h\right)^{2}-5\left(x+1\right)+1\right)-h\left(2x^{2}-5x+1\right)$$

Use the distributive property to multiply f by x+h.

$$fx+fh-fx=h\left(2\left(x+h\right)^{2}-5\left(x+1\right)+1\right)-h\left(2x^{2}-5x+1\right)$$

Combine fx and -fx to get 0.

$$fh=h\left(2\left(x+h\right)^{2}-5\left(x+1\right)+1\right)-h\left(2x^{2}-5x+1\right)$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+h\right)^{2}.

$$fh=h\left(2\left(x^{2}+2xh+h^{2}\right)-5\left(x+1\right)+1\right)-h\left(2x^{2}-5x+1\right)$$

Use the distributive property to multiply 2 by x^{2}+2xh+h^{2}.

$$fh=h\left(2x^{2}+4xh+2h^{2}-5\left(x+1\right)+1\right)-h\left(2x^{2}-5x+1\right)$$

Use the distributive property to multiply -5 by x+1.

$$fh=h\left(2x^{2}+4xh+2h^{2}-5x-5+1\right)-h\left(2x^{2}-5x+1\right)$$

Add -5 and 1 to get -4.

$$fh=h\left(2x^{2}+4xh+2h^{2}-5x-4\right)-h\left(2x^{2}-5x+1\right)$$

Use the distributive property to multiply h by 2x^{2}+4xh+2h^{2}-5x-4.

$$fh=2hx^{2}+4xh^{2}+2h^{3}-5hx-4h-h\left(2x^{2}-5x+1\right)$$

Use the distributive property to multiply h by 2x^{2}-5x+1.

$$fh=2hx^{2}+4xh^{2}+2h^{3}-5hx-4h-\left(2hx^{2}-5hx+h\right)$$

To find the opposite of 2hx^{2}-5hx+h, find the opposite of each term.

$$fh=2hx^{2}+4xh^{2}+2h^{3}-5hx-4h-2hx^{2}+5hx-h$$

Combine 2hx^{2} and -2hx^{2} to get 0.

$$fh=4xh^{2}+2h^{3}-5hx-4h+5hx-h$$

Combine -5hx and 5hx to get 0.

$$fh=4xh^{2}+2h^{3}-4h-h$$

Combine -4h and -h to get -5h.

$$fh=4xh^{2}+2h^{3}-5h$$

The equation is in standard form.

$$hf=4xh^{2}+2h^{3}-5h$$

Divide both sides by h.

$$\frac{hf}{h}=\frac{h\left(4hx+2h^{2}-5\right)}{h}$$

Dividing by h undoes the multiplication by h.

$$f=\frac{h\left(4hx+2h^{2}-5\right)}{h}$$

Divide h\left(4xh+2h^{2}-5\right) by h.

$$f=4hx+2h^{2}-5$$

Solve for h (complex solution)

$$\left\{\begin{matrix}h=\frac{\sqrt{4x^{2}+2f+10}}{2}-x\text{, }&\left(arg(x)\geq \pi \text{ and }x\neq 0\right)\text{ or }f\neq -5\\h=-\frac{\sqrt{4x^{2}+2f+10}}{2}-x\text{, }&\left(arg(x)<\pi \text{ and }x\neq 0\right)\text{ or }f\neq -5\end{matrix}\right.$$

Solve for h

$$\left\{\begin{matrix}h=\frac{\sqrt{4x^{2}+2f+10}}{2}-x\text{, }&\left(f\neq -5\text{ or }x<0\right)\text{ and }f\geq -2x^{2}-5\\h=-\frac{\sqrt{4x^{2}+2f+10}}{2}-x\text{, }&\left(f\neq -5\text{ or }x>0\right)\text{ and }f\geq -2x^{2}-5\end{matrix}\right.$$