Final answer:
To divide the polynomial as written, the numerator is first simplified to \(-48x - 36\), which does not result in a simple division by \(x - 4\). None of the answer choices A, B, C, or D match the result. However, if there was an error in the transcription of the polynomial and it should have been \(4x - 3\) instead of \(-36\), then the result would be \(-12\) after dividing by \(x - 4\).
Step-by-step explanation:
To divide the polynomial \(\frac{x+15x-77x+13x-36}{x-4}\) by \(x-4\), we must first simplify the numerator by combining like terms. The terms in the numerator are \(x\), \(15x\), \(-77x\), and \(13x\). When we combine these, we get:
\(x + 15x - 77x + 13x = 1x + 15x - 77x + 13x = (1+15-77+13)x = -48x\)
Thus, the simplified polynomial is \(\frac{-48x-36}{x-4}\). To divide \(-48x-36\) by \(x-4\), we can either use long division or factor out a \(-12\) from both terms in the numerator and simplify further:
\(-12(4x+3)\)
Since \(x - 4\) is the divisor, we can see that \(4x + 3\) is not divisible by \(x - 4\). However, if there was a mistake in the polynomial and it was meant to read \(4x - 3\) instead of \(-36\), then the division would result in \(-12\), as \(4x - 3\) would divide evenly by \(x - 4\). Given this, none of the answer choices provided (A, B, C, D) would be a result of the division as presented.