Sure, let's go through each part systematically to expand and simplify the given expressions.
### Part (1)
Expression: [tex]\( a \cdot -3q(-5 + 2q) \)[/tex]
Step 1: Distribute [tex]\(-3q \)[/tex] through the parentheses:
[tex]\[ -3q \cdot (-5) + -3q \cdot (2q) = 15q - 6q^2 \][/tex]
Step 2: Multiply by [tex]\(a\)[/tex]:
[tex]\[ a \cdot (15q - 6q^2) = 15aq - 6aq^2 \][/tex]
Therefore, the expanded and simplified expression for Part (1) is:
[tex]\[ -6aq^2 + 15aq \][/tex]
### Part (2)
Expression: [tex]\( 3a(x + 4a) - x(a - 4) \)[/tex]
Step 1: Distribute [tex]\(3a\)[/tex] through the first parentheses:
[tex]\[ 3a \cdot x + 3a \cdot 4a = 3ax + 12a^2 \][/tex]
Step 2: Distribute [tex]\(x\)[/tex] through the second parentheses:
[tex]\[ x \cdot a - x \cdot 4 = ax - 4x \][/tex]
Step 3: Combine the two results:
[tex]\[ 3ax + 12a^2 - ax + 4x \][/tex]
Step 4: Combine like terms:
[tex]\[ (3ax - ax) + 12a^2 + 4x = 2ax + 12a^2 + 4x \][/tex]
Therefore, the expanded and simplified expression for Part (2) is:
[tex]\[ 12a^2 + 2ax + 4x \][/tex]
### Part (3)
Expression: [tex]\( x(p - 3) + 2(3xp + 2x) - 5x(p + 4) \)[/tex]
Step 1: Distribute [tex]\(x\)[/tex] through the first parentheses:
[tex]\[ x \cdot p - x \cdot 3 = xp - 3x \][/tex]
Step 2: Distribute [tex]\(2\)[/tex] through the second parentheses:
[tex]\[ 2 \cdot 3xp + 2 \cdot 2x = 6xp + 4x \][/tex]
Step 3: Distribute [tex]\(-5x\)[/tex] through the third parentheses:
[tex]\[ -5x \cdot p - 5x \cdot 4 = -5xp - 20x \][/tex]
Step 4: Combine the results:
[tex]\[ xp - 3x + 6xp + 4x - 5xp - 20x \][/tex]
Step 5: Combine like terms:
[tex]\[ (xp + 6xp - 5xp) + (-3x + 4x - 20x) = 2xp - 19x \][/tex]
Therefore, the expanded and simplified expression for Part (3) is:
[tex]\[ 2px - 19x \][/tex]
### Summary
Combining all of our results:
1. [tex]\( -6aq^2 + 15aq \)[/tex]
2. [tex]\( 12a^2 + 2ax + 4x \)[/tex]
3. [tex]\( 2px - 19x \)[/tex]
