Multiplying Binomials Worksheet (pdf) and Answer Key. 29 ... - Free Printable

It seems you've uploaded a task involving the expansion of algebraic expressions. The problem involves multiplying binomials and trinomials. Let's solve each expression step by step.

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Problem 19: Expand \((9x - 4)(x - 6)\)

We use the distributive property (also known as the FOIL method for binomials):

\[
(9x - 4)(x - 6) = 9x \cdot x + 9x \cdot (-6) + (-4) \cdot x + (-4) \cdot (-6)
\]

Now, calculate each term:

1. \(9x \cdot x = 9x^2\)
2. \(9x \cdot (-6) = -54x\)
3. \((-4) \cdot x = -4x\)
4. \((-4) \cdot (-6) = 24\)

Combine all the terms:

\[
9x^2 - 54x - 4x + 24 = 9x^2 - 58x + 24
\]

So, the expanded form is:

\[
\boxed{9x^2 - 58x + 24}
\]

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Problem 21: Expand \((-5x + 6)(x + 5)\)

Again, use the distributive property:

\[
(-5x + 6)(x + 5) = -5x \cdot x + (-5x) \cdot 5 + 6 \cdot x + 6 \cdot 5
\]

Now, calculate each term:

1. \(-5x \cdot x = -5x^2\)
2. \((-5x) \cdot 5 = -25x\)
3. \(6 \cdot x = 6x\)
4. \(6 \cdot 5 = 30\)

Combine all the terms:

\[
-5x^2 - 25x + 6x + 30 = -5x^2 - 19x + 30
\]

So, the expanded form is:

\[
\boxed{-5x^2 - 19x + 30}
\]

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Problem 23: Expand \((x^2 - 8)(x + 4)\)

Use the distributive property:

\[
(x^2 - 8)(x + 4) = x^2 \cdot x + x^2 \cdot 4 + (-8) \cdot x + (-8) \cdot 4
\]

Now, calculate each term:

1. \(x^2 \cdot x = x^3\)
2. \(x^2 \cdot 4 = 4x^2\)
3. \((-8) \cdot x = -8x\)
4. \((-8) \cdot 4 = -32\)

Combine all the terms:

\[
x^3 + 4x^2 - 8x - 32
\]

So, the expanded form is:

\[
\boxed{x^3 + 4x^2 - 8x - 32}
\]

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Problem 25: Expand \((x^2 + 10x)(x - 3)\)

Use the distributive property:

\[
(x^2 + 10x)(x - 3) = x^2 \cdot x + x^2 \cdot (-3) + 10x \cdot x + 10x \cdot (-3)
\]

Now, calculate each term:

1. \(x^2 \cdot x = x^3\)
2. \(x^2 \cdot (-3) = -3x^2\)
3. \(10x \cdot x = 10x^2\)
4. \(10x \cdot (-3) = -30x\)

Combine all the terms:

\[
x^3 - 3x^2 + 10x^2 - 30x = x^3 + 7x^2 - 30x
\]

So, the expanded form is:

\[
\boxed{x^3 + 7x^2 - 30x}
\]

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Final Answers:

19. \(\boxed{9x^2 - 58x + 24}\)
21. \(\boxed{-5x^2 - 19x + 30}\)
23. \(\boxed{x^3 + 4x^2 - 8x - 32}\)
25. \(\boxed{x^3 + 7x^2 - 30x}\)