Determine the end behavior of the graph of the function. $$ p(x)=-7 x^{2}(4-x)(4 x-5)^{3} $$

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1. We start with the function
   $$
   p(x) = -7x^2(4-x)(4x-5)^3.
   $$

2. Determine the degree of the polynomial by adding the degrees from each factor:
   - $$x^2$$ has degree $$2$$,
   - $$4-x$$ has degree $$1$$,
   - $$(4x-5)^3$$ has degree $$3$$.
   
   Thus, the total degree is
   $$
   2+1+3=6.
   $$

3. Find the leading coefficient. We identify the contribution from each factor:
   - From $$x^2$$: the leading term is $$x^2$$.
   - From $$(4-x)$$: the leading term is $$-x$$ (since the highest power version is $$-x$$ when written in descending order).
   - From $$(4x-5)^3$$: the leading term is $$(4x)^3=64x^3$$.
   
   Multiplying these together with the constant $$-7$$, we have
   $$
   -7 \cdot x^2 \cdot (-x) \cdot (64x^3) = -7 \cdot (-1) \cdot 64 \cdot x^{2+1+3} = 448x^6.
   $$
   
   So the leading coefficient is $$448$$ (which is positive).

4. For a polynomial of even degree with a positive leading coefficient:
   - As $$x\to \infty$$, $$p(x)\to \infty$$.
   - As $$x\to -\infty$$, $$p(x)\to \infty$$.

Thus, the end behavior of the function is such that the graph rises to infinity on both ends.
As $$x$$ approaches positive or negative infinity, the graph of $$p(x) = -7x^2(4-x)(4x-5)^3$$ rises to infinity.

Determine the end behavior of the graph of the function.

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