Question:
Simplify this expression: (32*y + 13)*(18*y**2 + 24)
When performing calculations, please follow these guidelines:
1. Use ** instead of ^ to represent exponents. For example, write 7*X**2 instead of 7*X^2.
2. Always include the * symbol for all multiplication operations in your reasoning steps. For example, write `-3*X**3*sin(X) - 9*X**2*cos(X) + 18*X*sin(X) + 18*cos(X) + C` instead of `-3x3sin(x) - 9x2cos(x) + 18xsin(x) + 18cos(x) + C`.

Answer: 576*y**3 + 234*y**2 + 768*y + 312

Let's solve the polynomial multiplication step by step. We can expand all partial products first, then add all of them together. Here's the solution steps:
#Step 1:
Current expression: (32*y + 13)*(18*y**2 + 24)
Expanding the part (32*y + 13) using the distributive property, and we got: (32*y)*(18*y**2 + 24) + 13*(18*y**2 + 24)
<replace-#Step 1>#Step 2:
Current expression: (32*y)*(18*y**2 + 24) + 13*(18*y**2 + 24)</replace>
Simplifying the partial expression: 13*(18*y**2 + 24)
Expanding the part (18*y**2 + 24) using the distributive property, and we got: (18*y**2)*13 + 24*13
Replacing 13*(18*y**2 + 24) with (18*y**2)*13 + 24*13 in the overall expression.
<replace-#Step 2>#Step 3:
Current expression: (32*y)*(18*y**2 + 24) + (18*y**2)*13 + 312</replace>
Simplifying the partial expression: (32*y)*(18*y**2 + 24)
Expanding the part (18*y**2 + 24) using the distributive property, and we got: 24*(32*y) + (32*y)*(18*y**2)
Replacing (32*y)*(18*y**2 + 24) with 24*(32*y) + (32*y)*(18*y**2) in the overall expression.
<replace-#Step 3>#Step 4:
Current expression: 24*32*y + (32*y)*(18*y**2) + (18*y**2)*13 + 312</replace>
No partial expression needs using the distributive property to simplify.
Calculating the product of the terms in the current expression:
  (18*y**2)*13 = 234*y**2
  (18*y**2)*(32*y) = 576*y**3
  24*32*y = 768*y
Replacing them in the overall expression.
<replace-#Step 4>#Step 5:
Current expression: 576*y**3 + 234*y**2 + 768*y + 312</replace>
No more calculations needed.
<answer>576*y**3 + 234*y**2 + 768*y + 312</answer>


Question:
Calculate the following: (12 - 20*y)*(-y**3 - 48*y**2 + 40*y)
When performing calculations, please follow these guidelines:
1. Use ** instead of ^ to represent exponents. For example, write 7*X**2 instead of 7*X^2.
2. Always include the * symbol for all multiplication operations in your reasoning steps. For example, write `-3*X**3*sin(X) - 9*X**2*cos(X) + 18*X*sin(X) + 18*cos(X) + C` instead of `-3x3sin(x) - 9x2cos(x) + 18xsin(x) + 18cos(x) + C`.

Answer: 20*y**4 + 948*y**3 - 1376*y**2 + 480*y

Let's solve the polynomial multiplication step by step. We can expand all partial products first, then add all of them together. Here's the solution steps:
#Step 1:
Current expression: (12 - 20*y)*(-y**3 - 48*y**2 + 40*y)
Expanding the part (12 - 20*y) using the distributive property, and we got: (-20*y)*(-y**3 - 48*y**2 + 40*y) + 12*(-y**3 - 48*y**2 + 40*y)
<replace-#Step 1>#Step 2:
Current expression: (-20*y)*(-y**3 - 48*y**2 + 40*y) + 12*(-y**3 - 48*y**2 + 40*y)</replace>
Simplifying the partial expression: 12*(-y**3 - 48*y**2 + 40*y)
Expanding the part (-y**3 - 48*y**2 + 40*y) using the distributive property, and we got: (40*y)*12 - 48*y**2*12 - y**3*12
Replacing 12*(-y**3 - 48*y**2 + 40*y) with (40*y)*12 - 48*y**2*12 - y**3*12 in the overall expression.
<replace-#Step 2>#Step 3:
Current expression: (-20*y)*(-y**3 - 48*y**2 + 40*y) + (40*y)*12 - 48*y**2*12 - y**3*12</replace>
Simplifying the partial expression: (-20*y)*(-y**3 - 48*y**2 + 40*y)
Expanding the part (-y**3 - 48*y**2 + 40*y) using the distributive property, and we got: (-20*y)*(40*y) + (-20*y)*(-48*y**2) + (-20*y)*(-y**3)
Replacing (-20*y)*(-y**3 - 48*y**2 + 40*y) with (-20*y)*(40*y) + (-20*y)*(-48*y**2) + (-20*y)*(-y**3) in the overall expression.
<replace-#Step 3>#Step 4:
Current expression: (-20*y)*(40*y) + (-20*y)*(-48*y**2) + (-20*y)*(-y**3) + (40*y)*12 - 48*y**2*12 - y**3*12</replace>
No partial expression needs using the distributive property to simplify.
Calculating the product of the terms in the current expression:
  (-y**3)*12 = -12*y**3
  (-y**3)*(-20*y) = 20*y**4
  (-48*y**2)*12 = -576*y**2
  (-48*y**2)*(-20*y) = 960*y**3
  (40*y)*12 = 480*y
  (40*y)*(-20*y) = -800*y**2
Replacing them in the overall expression.
<replace-#Step 4>#Step 5:
Current expression: 20*y**4 - 12*y**3 + 960*y**3 - 800*y**2 - 576*y**2 + 480*y</replace>
Combining like terms, got 20*y**4 + 948*y**3 - 1376*y**2 + 480*y
<replace-#Step 5>#Step 6:
Current expression: 20*y**4 + 948*y**3 - 1376*y**2 + 480*y</replace>
No more calculations needed.
<answer>20*y**4 + 948*y**3 - 1376*y**2 + 480*y</answer>


Question:
Simplify this expression: (66 - 18*y)*(34*y**3 + 27)
When performing calculations, please follow these guidelines:
1. Use ** instead of ^ to represent exponents. For example, write 7*X**2 instead of 7*X^2.
2. Always include the * symbol for all multiplication operations in your reasoning steps. For example, write `-3*X**3*sin(X) - 9*X**2*cos(X) + 18*X*sin(X) + 18*cos(X) + C` instead of `-3x3sin(x) - 9x2cos(x) + 18xsin(x) + 18cos(x) + C`.

Answer: -612*y**4 + 2244*y**3 - 486*y + 1782

Let's solve the polynomial multiplication step by step. We can expand all partial products first, then add all of them together. Here's the solution steps:
#Step 1:
Current expression: (66 - 18*y)*(34*y**3 + 27)
Expanding the part (34*y**3 + 27) using the distributive property, and we got: (34*y**3)*(66 - 18*y) + 27*(66 - 18*y)
<replace-#Step 1>#Step 2:
Current expression: (34*y**3)*(66 - 18*y) + 27*(66 - 18*y)</replace>
Simplifying the partial expression: 27*(66 - 18*y)
Expanding the part (66 - 18*y) using the distributive property, and we got: -18*y*27 + 66*27
Replacing 27*(66 - 18*y) with -18*y*27 + 66*27 in the overall expression.
<replace-#Step 2>#Step 3:
Current expression: -18*y*27 + (34*y**3)*(66 - 18*y) + 1782</replace>
Simplifying the partial expression: (34*y**3)*(66 - 18*y)
Expanding the part (66 - 18*y) using the distributive property, and we got: (-18*y)*(34*y**3) + 66*(34*y**3)
Replacing (34*y**3)*(66 - 18*y) with (-18*y)*(34*y**3) + 66*(34*y**3) in the overall expression.
<replace-#Step 3>#Step 4:
Current expression: (-18*y)*(34*y**3) - 18*y*27 + 66*34*y**3 + 1782</replace>
No partial expression needs using the distributive property to simplify.
Calculating the product of the terms in the current expression:
  (-18*y)*27 = -486*y
  (-18*y)*(34*y**3) = -612*y**4
  66*34*y**3 = 2244*y**3
Replacing them in the overall expression.
<replace-#Step 4>#Step 5:
Current expression: -612*y**4 + 2244*y**3 - 486*y + 1782</replace>
No more calculations needed.
<answer>-612*y**4 + 2244*y**3 - 486*y + 1782</answer>