Divide (x^4+x^3+5x^2-10x+10)/(x^2+2x+1)

$\frac{x^{4} + x^{3} + 5 x^{2} - 10 x + 10}{x^{2} + 2 x + 1}$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$\frac{x^{4} + x^{3} + 5 x^{2} - 10 x + 10}{x^{2} + 2 x + 1^{2}}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

$\frac{x^{4} + x^{3} + 5 x^{2} - 10 x + 10}{x^{2} + 2 \cdot x \cdot 1 + 1^{2}}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\frac{x^{4} + x^{3} + 5 x^{2} - 10 x + 10}{{(x + 1)}^{2}}$

Split the fraction $\frac{x^{4} + x^{3} + 5 x^{2} - 10 x + 10}{{(x + 1)}^{2}}$ into two fractions.

$\frac{x^{4} + x^{3} + 5 x^{2} - 10 x}{{(x + 1)}^{2}} + \frac{10}{{(x + 1)}^{2}}$

Split the fraction $\frac{x^{4} + x^{3} + 5 x^{2} - 10 x}{{(x + 1)}^{2}}$ into two fractions.

$\frac{x^{4} + x^{3} + 5 x^{2}}{{(x + 1)}^{2}} + \frac{- 10 x}{{(x + 1)}^{2}} + \frac{10}{{(x + 1)}^{2}}$

Split the fraction $\frac{x^{4} + x^{3} + 5 x^{2}}{{(x + 1)}^{2}}$ into two fractions.

$\frac{x^{4} + x^{3}}{{(x + 1)}^{2}} + \frac{5 x^{2}}{{(x + 1)}^{2}} + \frac{- 10 x}{{(x + 1)}^{2}} + \frac{10}{{(x + 1)}^{2}}$

Split the fraction $\frac{x^{4} + x^{3}}{{(x + 1)}^{2}}$ into two fractions.

$\frac{x^{4}}{{(x + 1)}^{2}} + \frac{x^{3}}{{(x + 1)}^{2}} + \frac{5 x^{2}}{{(x + 1)}^{2}} + \frac{- 10 x}{{(x + 1)}^{2}} + \frac{10}{{(x + 1)}^{2}}$

Move the negative in front of the fraction.

$\frac{x^{4}}{{(x + 1)}^{2}} + \frac{x^{3}}{{(x + 1)}^{2}} + \frac{5 x^{2}}{{(x + 1)}^{2}} - \frac{10 x}{{(x + 1)}^{2}} + \frac{10}{{(x + 1)}^{2}}$