Question

divide. if there is no remainder, enter the quotient. if there is a remainder, enter your answer as quotient$+\frac{\text{remainder}}{\text{divisor}}$.
$left(5x^{3}-4x^{2}-5x - 18
ight)divleft(x - 2
ight)$

Explanation:

Step1: Set up polynomial long division

Divide $5x^3 - 4x^2 - 5x - 18$ by $x-2$. First term: $\frac{5x^3}{x}=5x^2$
Multiply divisor by $5x^2$: $5x^2(x-2)=5x^3-10x^2$
Subtract from dividend: $(5x^3-4x^2)-(5x^3-10x^2)=6x^2$
Bring down $-5x$: $6x^2-5x$

Step2: Calculate next term of quotient

Next term: $\frac{6x^2}{x}=6x$
Multiply divisor by $6x$: $6x(x-2)=6x^2-12x$
Subtract: $(6x^2-5x)-(6x^2-12x)=7x$
Bring down $-18$: $7x-18$

Step3: Calculate final quotient term and remainder

Final quotient term: $\frac{7x}{x}=7$
Multiply divisor by $7$: $7(x-2)=7x-14$
Subtract: $(7x-18)-(7x-14)=-4$

Answer:

$5x^2 + 6x + 7 + \frac{-4}{x-2}$