To solve the given division [tex]\( \frac{4x^3 - 14x^2 - 7x - 4}{x - 4} \)[/tex]:
1. We start by identifying the polynomial in the numerator and the linear polynomial in the denominator.
2. We perform polynomial long division to divide the numerator by the denominator.
After performing polynomial division, we find that:
[tex]\[ \frac{4x^3 - 14x^2 - 7x - 4}{x - 4} \][/tex]
can be expressed as:
[tex]\[ 4x^2 + 2x + 1 \][/tex]
Therefore, the quotient is [tex]\(4x^2 + 2x + 1\)[/tex], and the remainder is [tex]\(0\)[/tex]. This means that the division is exact and there are no additional terms involving [tex]\( \frac{k}{x-4} \)[/tex].
So, the final answer is:
[tex]\[ \frac{4x^3 - 14x^2 - 7x - 4}{x - 4} = 4x^2 + 2x + 1 \][/tex]
