Find the coefficient of \(\displaystyle x^5\) in \(\displaystyle (x^4 + 4)(3x-4)^9\)

**My solution**

So I started off by separating both brackets into 2 separate equations.

1. \(\displaystyle x^4(3x-4)^9\)

2. \(\displaystyle 4(3x-4)^9\)

**Now I find the coefficient of**\(\displaystyle x\)

**in equation 1 and I find the coefficient of**\(\displaystyle x^5\)

**in equation 2.**

So: \(\displaystyle \binom{9}{9-i}\) \(\displaystyle (3x)^{9-i}\) \(\displaystyle (-4)^i\)

now:

1. \(\displaystyle \binom{9}{8}\) \(\displaystyle (3)^8\) \(\displaystyle (-4)^1\) = \(\displaystyle 9 \times 6561 \times (-4)\)

2. \(\displaystyle 4 \binom{9}{4}\) \(\displaystyle (3)^4\) \(\displaystyle (-4)^5\) = \(\displaystyle 4 \times 126 \times 81 \times (-1024)\)

So now adding both up:

Solution: \(\displaystyle 9 \times 6561 \times (-4) + 4 \times 126 \times 81 \times (-1024)\)