Answer :

To divide
[tex]$$12x^3 - 19x^2 - 25x - 10$$[/tex]
by
[tex]$$4x + 3,$$[/tex]
we perform polynomial long division.

1. First, divide the leading term of the numerator, [tex]$12x^3$[/tex], by the leading term of the divisor, [tex]$4x$[/tex]. This gives:
[tex]$$
\frac{12x^3}{4x} = 3x^2.
$$[/tex]

2. Multiply the entire divisor by [tex]$3x^2$[/tex]:
[tex]$$
3x^2(4x + 3) = 12x^3 + 9x^2.
$$[/tex]

3. Subtract this from the original numerator:
[tex]$$
\begin{aligned}
(12x^3 - 19x^2) - (12x^3 + 9x^2) &= 12x^3 - 19x^2 - 12x^3 - 9x^2 \\
&= -28x^2.
\end{aligned}
$$[/tex]
Bring down the next term, [tex]$-25x$[/tex], to obtain:
[tex]$$
-28x^2 - 25x.
$$[/tex]

4. Next, divide [tex]$-28x^2$[/tex] by the leading term of the divisor [tex]$4x$[/tex]:
[tex]$$
\frac{-28x^2}{4x} = -7x.
$$[/tex]

5. Multiply the divisor by [tex]$-7x$[/tex]:
[tex]$$
-7x(4x + 3) = -28x^2 - 21x.
$$[/tex]

6. Subtract this from the expression obtained in step 3:
[tex]$$
\begin{aligned}
(-28x^2 - 25x) - (-28x^2 - 21x) &= -28x^2 - 25x + 28x^2 + 21x \\
&= -4x.
\end{aligned}
$$[/tex]
Bring down the next term, [tex]$-10$[/tex], to get:
[tex]$$
-4x - 10.
$$[/tex]

7. Finally, divide [tex]$-4x$[/tex] by [tex]$4x$[/tex]:
[tex]$$
\frac{-4x}{4x} = -1.
$$[/tex]

8. Multiply the divisor by [tex]$-1$[/tex]:
[tex]$$
-1(4x + 3) = -4x - 3.
$$[/tex]

9. Subtract this from the expression obtained in step 6:
[tex]$$
\begin{aligned}
(-4x - 10) - (-4x - 3) &= -4x - 10 + 4x + 3 \\
&= -7.
\end{aligned}
$$[/tex]

The division yields a quotient of
[tex]$$3x^2 - 7x - 1$$[/tex]
with a remainder of [tex]$-7$[/tex]. Therefore, the result of the division is expressed as:
[tex]$$
\frac{12x^3 - 19x^2 - 25x - 10}{4x + 3} = 3x^2 - 7x - 1 + \frac{-7}{4x+3}.
$$[/tex]

This is the complete step-by-step solution.

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