Which of the following is equivalent to \$(x^2 + 3x + 7)/(x + 4)\$.

To divide \$x^2+ 3x + 7\$ by x + 4, set up a long division problem.

Divide the leading term of the dividend by the leading term of the divisor:
⇒ Divide \$x^2\$ by x to get x.

⇒ Write x above the division bar (as part of your quotient).

\$ x^2/x = x\$

Multiply the entire divisor x + 4 by x (your current quotient):

⇒ Write this result under the dividend

\$x \times (x + 4) = (x^2 + 4x)\$

Subtract \$x^2 + 4x\$ from \$x^2 + 3x + 7\$:

⇒ Subtract term by term:

⇒ Bring down the resulting -x + 7

\$ x^2 - x^2 = 0\$
3x - 4x = - x
The constant remains + 7.

Divide the leading term of the new dividend −x + 7 by the leading term of the divisor:
⇒ Divide - x by x to get - 1

⇒ Add - 1 to the quotient above the division bar

\$-x/x\$ = - 1

Multiply x + 4 by - 1:

⇒ Write this result under the -x + 7

\$-1 \times (x + 4) = -x - 4\$

Subtract -x - 4 from -x + 7:

-x + x = 0
7 - (- 4) = (7 + 4) = 11
This 11 is the remainder.

Final Quotient and Remainder

Quotient: x − 1
Remainder: 11

So, the result of the division \$(x^2 + 3x + 7)/(x + 4)\$ is (x - 1 + \$11/(x+4)\$).

This option simplifies to \$10/4\$ = 2.5, which does not represent the original rational function involving

\$(3 + 7)/4\$

This is a linear expression with a constant fraction, which doesn’t match the quotient plus a term involving the variable in the denominator

\$x + (3/4)\$

This option retains a dependence on x in a fractional part, but the linear part and the constants do not align with the division result

\$3 + (7/(x+4))\$

This is exactly the expression we derived from the polynomial long division: the polynomial part x - 1 plus a fractional remainder \$ 11/(x+4)\$

\$x - 1 + 11/(x+4)\$

Option

D

Can you summarize what you’ve understood in the above steps?