Certainly! Let's perform the polynomial long division to divide [tex]\( 15x^2 + 14x - 8 \)[/tex] by [tex]\( 3x + 4 \)[/tex].
Step 1: Setup the Division
We start by writing the dividend [tex]\( 15x^2 + 14x - 8 \)[/tex] and the divisor [tex]\( 3x + 4 \)[/tex]:
[tex]\[
\frac{15x^2 + 14x - 8}{3x + 4}
\][/tex]
Step 2: Divide the Leading Terms
Divide the leading term of the dividend [tex]\( 15x^2 \)[/tex] by the leading term of the divisor [tex]\( 3x \)[/tex]:
[tex]\[
\frac{15x^2}{3x} = 5x
\][/tex]
So, the first term of the quotient is [tex]\( 5x \)[/tex].
Step 3: Multiply and Subtract
Multiply [tex]\( 5x \)[/tex] by the divisor [tex]\( 3x + 4 \)[/tex]:
[tex]\[
5x \cdot (3x + 4) = 15x^2 + 20x
\][/tex]
Subtract this from the original dividend:
[tex]\[
(15x^2 + 14x - 8) - (15x^2 + 20x) = (14x - 20x) - 8 = -6x - 8
\][/tex]
Step 4: Divide the Remaining Terms
Now, divide the leading term [tex]\( -6x \)[/tex] by the leading term [tex]\( 3x \)[/tex]:
[tex]\[
\frac{-6x}{3x} = -2
\][/tex]
So, the next term of the quotient is [tex]\( -2 \)[/tex].
Step 5: Multiply and Subtract
Multiply [tex]\( -2 \)[/tex] by the divisor [tex]\( 3x + 4 \)[/tex]:
[tex]\[
-2 \cdot (3x + 4) = -6x - 8
\][/tex]
Subtract this from the current remainder:
[tex]\[
(-6x - 8) - (-6x - 8) = 0
\][/tex]
Since the remainder is 0, we have completed the division.
Step 6: Write the Quotient
The quotient of the polynomial division is:
[tex]\[
5x - 2
\][/tex]
Final Answer:
The quotient of dividing [tex]\( 15x^2 + 14x - 8 \)[/tex] by [tex]\( 3x + 4 \)[/tex] is [tex]\( 5x - 2 \)[/tex].
Therefore, the correct choice is:
[tex]\[ \boxed{5x - 2} \][/tex]
