Answer:
![\large\boxed{1.\ V=3b^4-15b^3+3b^2-15b}\\\boxed{2.\ V=4\pi n^3-16\pi n^2+16\pi n}\\\boxed{3.\ V=2k^3-k^2-6k}](https://tex.z-dn.net/?f=%5Clarge%5Cboxed%7B1.%5C%20V%3D3b%5E4-15b%5E3%2B3b%5E2-15b%7D%5C%5C%5Cboxed%7B2.%5C%20V%3D4%5Cpi%20n%5E3-16%5Cpi%20n%5E2%2B16%5Cpi%20n%7D%5C%5C%5Cboxed%7B3.%5C%20V%3D2k%5E3-k%5E2-6k%7D)
Step-by-step explanation:
<h3>The picture #1:</h3>
It's a rectangular prism. The formula of a volume of a rectangular prism:
![V=lwh](https://tex.z-dn.net/?f=V%3Dlwh)
l - length
w - width
h - height
We have
![l=b+1,\ w=3b,\ h=b-5](https://tex.z-dn.net/?f=l%3Db%2B1%2C%5C%20w%3D3b%2C%5C%20h%3Db-5)
Substitute:
![V=(b+1)(3b)(b-5)](https://tex.z-dn.net/?f=V%3D%28b%2B1%29%283b%29%28b-5%29)
<em>Use the distributive property </em><em>a(b + c) = ab + ac</em>
<em>and the FOIL:</em><em> (a + b)(c + d) = ac + ad + bc + bd</em>
![V=(3b^3+3b)(b-5)=3b^4-15b^3+3b^2-15b](https://tex.z-dn.net/?f=V%3D%283b%5E3%2B3b%29%28b-5%29%3D3b%5E4-15b%5E3%2B3b%5E2-15b)
<h3>The picture #2:</h3>
It's a cone. The fomula of a volume of a cone:
![V=\dfrac{1}{3}\pi r^2h](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%5Cpi%20r%5E2h)
r - radius
h - height
We have:
![r=n-2,\ h=12n](https://tex.z-dn.net/?f=r%3Dn-2%2C%5C%20h%3D12n)
Substitute:
![V=\dfrac{1}{3}\pi(n-2)^2(12n)](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%5Cpi%28n-2%29%5E2%2812n%29)
<em>Use </em><em>(a + b)² = a² + 2ab + b²</em>
![V=\dfrac{1}{3}\pi(n^2-4n+4)(12n)=(4\pi n)(n^2-4n+4)](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%5Cpi%28n%5E2-4n%2B4%29%2812n%29%3D%284%5Cpi%20n%29%28n%5E2-4n%2B4%29)
<em>Use the distributive property:</em>
![V=4\pi n^3-16\pi n^2+16\pi n](https://tex.z-dn.net/?f=V%3D4%5Cpi%20n%5E3-16%5Cpi%20n%5E2%2B16%5Cpi%20n)
<h3>The picture #3:</h3>
It's a pyramid with a rectangle in the base. The formula of a volume of a rectangular pyramid:
![V=\dfrac{1}{3}abh](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7Dabh)
a, b - edge of a base
h - height
We have
![a=k-2, b=2k+3, h=3k](https://tex.z-dn.net/?f=a%3Dk-2%2C%20b%3D2k%2B3%2C%20h%3D3k)
Substitute:
![V=\dfrac{1}{3}(k-2)(2k+3)(3k)](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%28k-2%29%282k%2B3%29%283k%29)
<em>Use the distributive property and the FOIL.</em>
![V=\dfrac{1}{3}(k-2)(6k^2+9k)=\dfrac{1}{3}(6k^3+9k^2-12k^2-18k)\\\\=\dfrac{1}{3}(6k^3-3k^2-18k)=2k^3-k^2-6k](https://tex.z-dn.net/?f=V%3D%5Cdfrac%7B1%7D%7B3%7D%28k-2%29%286k%5E2%2B9k%29%3D%5Cdfrac%7B1%7D%7B3%7D%286k%5E3%2B9k%5E2-12k%5E2-18k%29%5C%5C%5C%5C%3D%5Cdfrac%7B1%7D%7B3%7D%286k%5E3-3k%5E2-18k%29%3D2k%5E3-k%5E2-6k)