Polynomial long division practice problems with space for solutions.

Here are the step-by-step solutions for each polynomial long division problem.

1) $(3k^2 + 10k - 9) \div (k - 9)$

Step 1: Divide the first term of the dividend ($3k^2$) by the first term of the divisor ($k$).
$3k^2 \div k = 3k$. This is the first part of the answer.

Step 2: Multiply $3k$ by the entire divisor $(k - 9)$.
$3k(k - 9) = 3k^2 - 27k$.

Step 3: Subtract this result from the dividend. Be careful with signs:
$(3k^2 + 10k) - (3k^2 - 27k) = 10k + 27k = 37k$.
Bring down the next term, $-9$, to get $37k - 9$.

Step 4: Divide the new leading term ($37k$) by $k$.
$37k \div k = 37$. This is the second part of the answer.

Step 5: Multiply $37$ by $(k - 9)$.
$37(k - 9) = 37k - 333$.

Step 6: Subtract again.
$(37k - 9) - (37k - 333) = -9 + 333 = 324$.

The remainder is $324$. We write it as a fraction over the divisor.

Answer: $3k + 37 + \frac{324}{k - 9}$

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2) $(-3y^2 + 6) \div (y - 4)$

*Note: The dividend is missing the $y$ term, so we treat it as $-3y^2 + 0y + 6$.*

Step 1: Divide $-3y^2$ by $y$.
$-3y^2 \div y = -3y$.

Step 2: Multiply $-3y$ by $(y - 4)$.
$-3y(y - 4) = -3y^2 + 12y$.

Step 3: Subtract.
$(-3y^2 + 0y) - (-3y^2 + 12y) = -12y$.
Bring down the $+6$ to get $-12y + 6$.

Step 4: Divide $-12y$ by $y$.
$-12y \div y = -12$.

Step 5: Multiply $-12$ by $(y - 4)$.
$-12(y - 4) = -12y + 48$.

Step 6: Subtract.
$(-12y + 6) - (-12y + 48) = 6 - 48 = -42$.

The remainder is $-42$.

Answer: $-3y - 12 - \frac{42}{y - 4}$

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3) $(z^2 - 19z - 11) \div (z + 8)$

Step 1: Divide $z^2$ by $z$.
$z^2 \div z = z$.

Step 2: Multiply $z$ by $(z + 8)$.
$z(z + 8) = z^2 + 8z$.

Step 3: Subtract.
$(z^2 - 19z) - (z^2 + 8z) = -19z - 8z = -27z$.
Bring down $-11$ to get $-27z - 11$.

Step 4: Divide $-27z$ by $z$.
$-27z \div z = -27$.

Step 5: Multiply $-27$ by $(z + 8)$.
$-27(z + 8) = -27z - 216$.

Step 6: Subtract.
$(-27z - 11) - (-27z - 216) = -11 + 216 = 205$.

The remainder is $205$.

Answer: $z - 27 + \frac{205}{z + 8}$

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4) $(-b^2 + 16b^2 - 13b + 5) \div (b + 4)$

*First, simplify the dividend by combining like terms:*
$-b^2 + 16b^2 = 15b^2$.
So the problem is: $(15b^2 - 13b + 5) \div (b + 4)$.

Step 1: Divide $15b^2$ by $b$.
$15b^2 \div b = 15b$.

Step 2: Multiply $15b$ by $(b + 4)$.
$15b(b + 4) = 15b^2 + 60b$.

Step 3: Subtract.
$(15b^2 - 13b) - (15b^2 + 60b) = -13b - 60b = -73b$.
Bring down $+5$ to get $-73b + 5$.

Step 4: Divide $-73b$ by $b$.
$-73b \div b = -73$.

Step 5: Multiply $-73$ by $(b + 4)$.
$-73(b + 4) = -73b - 292$.

Step 6: Subtract.
$(-73b + 5) - (-73b - 292) = 5 + 292 = 297$.

The remainder is $297$.

Answer: $15b - 73 + \frac{297}{b + 4}$

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5) $(2h^2 + 13h^2 - 14h + 4) \div (h + 3)$

*First, simplify the dividend by combining like terms:*
$2h^2 + 13h^2 = 15h^2$.
So the problem is: $(15h^2 - 14h + 4) \div (h + 3)$.

Step 1: Divide $15h^2$ by $h$.
$15h^2 \div h = 15h$.

Step 2: Multiply $15h$ by $(h + 3)$.
$15h(h + 3) = 15h^2 + 45h$.

Step 3: Subtract.
$(15h^2 - 14h) - (15h^2 + 45h) = -14h - 45h = -59h$.
Bring down $+4$ to get $-59h + 4$.

Step 4: Divide $-59h$ by $h$.
$-59h \div h = -59$.

Step 5: Multiply $-59$ by $(h + 3)$.
$-59(h + 3) = -59h - 177$.

Step 6: Subtract.
$(-59h + 4) - (-59h - 177) = 4 + 177 = 181$.

The remainder is $181$.

Answer: $15h - 59 + \frac{181}{h + 3}$

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6) $(z^2 - 19z - 11) \div (z + 8)$

This is the exact same problem as number 3.

Answer: $z - 27 + \frac{205}{z + 8}$

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Final Answer:
1) $3k + 37 + \frac{324}{k - 9}$
2) $-3y - 12 - \frac{42}{y - 4}$
3) $z - 27 + \frac{205}{z + 8}$
4) $15b - 73 + \frac{297}{b + 4}$
5) $15h - 59 + \frac{181}{h + 3}$
6) $z - 27 + \frac{205}{z + 8}$