Question

divide using long division. check your answer.
$(4x^2 - 5x + 2) \div (x - 2)$
the quotient is \square with remainder \square.

Explanation:

Step1: Divide leading terms

$\frac{4x^2}{x} = 4x$
Multiply divisor by $4x$: $4x(x-2)=4x^2-8x$
Subtract from dividend:
$(4x^2-5x+2)-(4x^2-8x) = 3x+2$

Step2: Divide new leading terms

$\frac{3x}{x}=3$
Multiply divisor by 3: $3(x-2)=3x-6$
Subtract from new dividend:
$(3x+2)-(3x-6)=8$

Step3: Verify the result

Use formula: $\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}$
$(x-2)(4x+3)+8 = 4x^2+3x-8x-6+8=4x^2-5x+2$, which matches the original dividend.

Answer:

The quotient is $4x+3$ with remainder $8$.