Expand brackets and simplify

Predict the expanded product

The rectangle has side lengths x plus a and x plus b. The drawing holds x at 4 units. Change a and b to 1 and 4, then 4 and 1. Track the two strips and the corner; the algebra describes any positive x

Which rule matches the labelled areas for every slider position?

Distribute, then collect

Brackets group a sum or difference. An outside factor multiplies every term, including its sign

1. 1Distribute to every term Multiply the outside factor by every term inside its bracket \(\displaystyle 3(x+4)=3x+12\)
2. 2Carry each sign A negative factor multiplies both positive and negative terms \(\displaystyle -2(x-5)=-2x+10\)

Which expression comes from multiplying every term inside the bracket?

What constant term results after expansion?

Expand both brackets, then collect like terms

From areas to signed products

Positive lengths explain the area model. The distributive law also works with negative terms, where an ordinary positive-area drawing is insufficient

1. 1First pair Multiply two x by x \(\displaystyle (2x)x=2x^{2}\)
2. 2Second pair Multiply two x by negative four \(\displaystyle (2x)(-4)=-8x\)
3. 3Third pair Multiply three by x \(\displaystyle 3x\)
4. 4Fourth pair Multiply three by negative four \(\displaystyle 3(-4)=-12\)

For this product, what is the coefficient of x after collecting the two cross-products?

What constant term belongs to the same expanded product?

Choose the full expansion

A squared bracket means the same bracket multiplied by itself. Expand it

What happens when the two cross-products cancel?

Evaluate the product when x is negative two. Use expansion or direct substitution

Which expansion keeps both product signs correct?