Question

find the quotient and remainder using long division for: \\(\frac{2x^3 - 16x^2 + 7x - 34}{2x^2 + 5}\\)\
the quotient is \\(\boxed{}\\)\
the remainder is \\(\boxed{}\\)\
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Explanation:

Step1: Divide leading terms

$\frac{2x^3}{2x^2} = x$

Step2: Multiply divisor by $x$

$x(2x^2 + 5) = 2x^3 + 5x$

Step3: Subtract from dividend

$(2x^3 - 16x^2 + 7x - 34) - (2x^3 + 5x) = -16x^2 + 2x - 34$

Step4: Divide new leading terms

$\frac{-16x^2}{2x^2} = -8$

Step5: Multiply divisor by $-8$

$-8(2x^2 + 5) = -16x^2 - 40$

Step6: Subtract to find remainder

$(-16x^2 + 2x - 34) - (-16x^2 - 40) = 2x + 6$

Answer:

The quotient is $x - 8$
The remainder is $2x + 6$