Expand brackets and collect like terms

\(\displaystyle 6(q+5)\)

Which expanded form is equivalent to the given expression?

The bracket contains one pronumeral tile and two unit tiles. Filled tiles represent positive terms and hollow tiles represent negative terms. At zero, both counts are zero and there are no tiles. Move the outside factor to positive and negative settings. Compare both rows and their signed counts

When the outside factor is negative, which rows have hollow tiles?

Expand, then collect

The outside factor multiplies every term inside the bracket. Keep each term's sign. After expansion, use the like-term rule from the previous topic

1. 1Multiply every term Write both products before simplifying them \(\displaystyle \begin{aligned}5(3p+2)&=5(3p)+5(2)\\&=15p+10\end{aligned}\)
2. 2Keep the negative factor Use the signs of both factors in each product \(\displaystyle \begin{aligned}&-5(2t-7)\\&=(-5)(2t)+(-5)(-7)\\&=-10t+35\end{aligned}\)
3. 3Multiply the pronumeral factors too A monomial factor multiplies the coefficient and the pronumeral part of each term \(\displaystyle \begin{aligned}&4h(3h-5)\\&=(4h)(3h)+(4h)(-5)\\&=12h^2-20h\end{aligned}\)
4. 4Powers and two pronumerals Use the same multiplication rule for higher powers and different pronumerals. Add the exponents when multiplying powers of the same pronumeral \(\displaystyle \begin{aligned}&-2a^2(3a+4b)\\&=(-2a^2)(3a)+(-2a^2)(4b)\\&=-6a^3-8a^2b\end{aligned}\)
5. 5Keep the untouched terms Expand the bracket before collecting like terms. A term outside the bracket stays in the expression \(\displaystyle \begin{aligned}&8-3(2s-5)\\&=8+(-3)(2s)+(-3)(-5)\\&=8-6s+15\\&=-6s+23\end{aligned}\)

\(\displaystyle -4(3u-2)\)

Which expanded form is equivalent to the given expression?

Keep the negative outside factor in both products

\(\displaystyle -3r(2r+5)\)

Which expanded form is equivalent to the given expression?

Multiply the numerical factors and the pronumeral factors in each product

\(\displaystyle 7(2v-3)-3(v-4)\)

Expand and collect like terms. Which result is equivalent to the given expression?

Treat the factor before the second bracket as negative, then collect the matching terms

What caused the first error in the student's working?