Use long division to divide. Express any numbers as integers or simplified fractions $$ \left(7 x^{3}-2 x^{2}+5\right) \div(2 x+1)=\square $$

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We wish to divide
$$
7x^3 - 2x^2 + 5
$$
by
$$
2x+1.
$$
Since the dividend is missing an $$x$$–term, we rewrite it as
$$
7x^3 - 2x^2 + 0x + 5.
$$

**Step 1.** Divide the first term:

Divide the leading term of the dividend by the leading term of the divisor:
$$
\frac{7x^3}{2x} = \frac{7}{2}x^2.
$$
This is the first term of the quotient.

Now multiply the divisor by $$\frac{7}{2}x^2$$:
$$
\frac{7}{2}x^2(2x+1)= 7x^3 + \frac{7}{2}x^2.
$$

Subtract this from the dividend:
$$
\begin{array}{rcl}
\left(7x^3 - 2x^2\right) - \left(7x^3 + \frac{7}{2}x^2\right)
&=& 7x^3 - 2x^2 - 7x^3 - \frac{7}{2}x^2\[1mm]
&=& -2x^2 - \frac{7}{2}x^2\[1mm]
&=& -\frac{4}{2}x^2 - \frac{7}{2}x^2\[1mm]
&=& -\frac{11}{2}x^2.
\end{array}
$$
Bring down the next term $$0x$$ so we have:
$$
-\frac{11}{2}x^2 + 0x.
$$

**Step 2.** Divide the new leading term:

Divide
$$
-\frac{11}{2}x^2 \div 2x = -\frac{11}{4}x.
$$
This is the next term of the quotient.

Multiply the divisor by $$-\frac{11}{4}x$$:
$$
-\frac{11}{4}x(2x+1) = -\frac{11}{4}x\cdot 2x - \frac{11}{4}x = -\frac{11}{2}x^2 - \frac{11}{4}x.
$$

Subtract this product from the current expression:
$$
\begin{array}{rcl}
\left(-\frac{11}{2}x^2 + 0x\right) - \left(-\frac{11}{2}x^2 - \frac{11}{4}x\right)
&=& -\frac{11}{2}x^2 + \frac{11}{2}x^2 + \frac{11}{4}x\[1mm]
&=& \frac{11}{4}x.
\end{array}
$$
Bring down the next term $$+5$$ to obtain:
$$
\frac{11}{4}x + 5.
$$

**Step 3.** Divide the new leading term:

Divide
$$
\frac{11}{4}x \div 2x = \frac{11}{8}.
$$
This is the next term of the quotient.

Multiply the divisor by $$\frac{11}{8}$$:
$$
\frac{11}{8}(2x+1)= \frac{11}{8}\cdot 2x + \frac{11}{8} = \frac{11}{4}x + \frac{11}{8}.
$$

Subtract this from $$\frac{11}{4}x+5$$:
$$
\begin{array}{rcl}
\left(\frac{11}{4}x+5\right) - \left(\frac{11}{4}x + \frac{11}{8}\right)
&=& \frac{11}{4}x - \frac{11}{4}x + 5 - \frac{11}{8}\[1mm]
&=& 5 - \frac{11}{8}.
\end{array}
$$

To combine the numbers, express $$5$$ as $$\frac{40}{8}$$:
$$
5 - \frac{11}{8} = \frac{40}{8} - \frac{11}{8} = \frac{29}{8}.
$$
This is the remainder.

**Final Answer:**

The quotient is
$$
\frac{7}{2}x^2 - \frac{11}{4}x + \frac{11}{8}
$$
with a remainder of
$$
\frac{29}{8}.
$$
Thus, the division is expressed as:
$$
\frac{7x^3-2x^2+5}{2x+1}=\frac{7}{2}x^2-\frac{11}{4}x+\frac{11}{8}+\frac{\frac{29}{8}}{2x+1}.
$$
The division of $$7x^3 - 2x^2 + 5$$ by $$2x + 1$$ results in a quotient of $$\frac{7}{2}x^2 - \frac{11}{4}x + \frac{11}{8}$$ with a remainder of $$\frac{29}{8}$$.

Use long division to divide. Express any numbers as integers or simplified fractions \[ \left(7 x^{3}-2 x^{2}+5\right) \div(2 x+1)=\square \]

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