Distribute $2x$: $2x(x^2 - 4x + 5) = 2x^3 - 8x^2 + 10x$

Distribute $3$: $3(x^2 - 4x + 5) = 3x^2 - 12x + 15$

Combine: $2x^3 - 8x^2 + 10x + 3x^2 - 12x + 15$ $= 2x^3 - 5x^2 - 2x + 15$

Long Division of Polynomials

Similar to long division with numbers.

Steps:

1. Divide the leading terms
2. Multiply and subtract
3. Bring down the next term
4. Repeat until done

Synthetic Division

A shortcut for dividing by $(x - c)$.

Use only when divisor is in form $(x - c)$.

Remainder Theorem

When polynomial $P(x)$ is divided by $(x - c)$: $\text{Remainder} = P(c)$

Answer: Remainder = $2$

Step 2: $-x^2 \div x = -x$ Multiply: $-x(x + 2) = -x^2 - 2x$ Subtract: $(-x^2 - 5x) - (-x^2 - 2x) = -3x$

Step 3: $-3x \div x = -3$ Multiply: $-3(x + 2) = -3x - 6$ Subtract: $(-3x + 1) - (-3x - 6) = 7$

Answer: $2x^2 - x - 3 + \frac{7}{x + 2}$