Question

1. ((x^4 - x^2 - 7) div (x + 4))

Explanation:

Step1: Rewrite the dividend

We have the dividend \(x^4 - x^2 - 7\). Let's rewrite it as \(x^4+0x^3 - x^2+0x - 7\) to perform polynomial long division.

Step2: Divide the leading terms

Divide the leading term of the dividend \(x^4\) by the leading term of the divisor \(x\), we get \(x^3\). Multiply the divisor \((x + 4)\) by \(x^3\) to get \(x^4+4x^3\). Subtract this from the dividend:

$$
LATEXBLOCK0
$$

Step3: Divide the new leading term

Divide the leading term of the new dividend \(-4x^3\) by the leading term of the divisor \(x\), we get \(-4x^2\). Multiply the divisor \((x + 4)\) by \(-4x^2\) to get \(-4x^3-16x^2\). Subtract this from the new dividend:

$$
LATEXBLOCK1
$$

Step4: Divide the new leading term

Divide the leading term of the new dividend \(15x^2\) by the leading term of the divisor \(x\), we get \(15x\). Multiply the divisor \((x + 4)\) by \(15x\) to get \(15x^2+60x\). Subtract this from the new dividend:

$$
LATEXBLOCK2
$$

Step5: Divide the new leading term

Divide the leading term of the new dividend \(-60x\) by the leading term of the divisor \(x\), we get \(-60\). Multiply the divisor \((x + 4)\) by \(-60\) to get \(-60x-240\). Subtract this from the new dividend:

$$
LATEXBLOCK3
$$

Step6: Write the result

So, \(\frac{x^4 - x^2 - 7}{x + 4}=x^3-4x^2 + 15x-60+\frac{233}{x + 4}\)

Answer:

\(x^3-4x^2 + 15x-60+\frac{233}{x + 4}\)