Here are the step-by-step solutions for each problem. To solve these, we use the distributive property: multiply the term outside the parentheses by each term inside the parentheses.
1. $3p(-4p + 2)$
* Multiply $3p$ by $-4p$: $3 \cdot -4 = -12$, and $p \cdot p = p^2$. Result: $-12p^2$
* Multiply $3p$ by $2$: $3 \cdot 2 = 6$, keep the $p$. Result: $6p$
* Combine them:
$-12p^2 + 6p$
2. $-4a(4a - 5)$
* Multiply $-4a$ by $4a$: $-4 \cdot 4 = -16$, and $a \cdot a = a^2$. Result: $-16a^2$
* Multiply $-4a$ by $-5$: Negative times negative is positive. $4 \cdot 5 = 20$, keep the $a$. Result: $+20a$
* Combine them:
$-16a^2 + 20a$
3. $5w(-4w - 3)$
* Multiply $5w$ by $-4w$: $5 \cdot -4 = -20$, and $w \cdot w = w^2$. Result: $-20w^2$
* Multiply $5w$ by $-3$: $5 \cdot -3 = -15$, keep the $w$. Result: $-15w$
* Combine them:
$-20w^2 - 15w$
4. $6b(-2b - 5)$
* Multiply $6b$ by $-2b$: $6 \cdot -2 = -12$, and $b \cdot b = b^2$. Result: $-12b^2$
* Multiply $6b$ by $-5$: $6 \cdot -5 = -30$, keep the $b$. Result: $-30b$
* Combine them:
$-12b^2 - 30b$
5. $-2x(-5x - 1)$
* Multiply $-2x$ by $-5x$: Negative times negative is positive. $2 \cdot 5 = 10$, and $x \cdot x = x^2$. Result: $10x^2$
* Multiply $-2x$ by $-1$: Negative times negative is positive. $2 \cdot 1 = 2$, keep the $x$. Result: $+2x$
* Combine them:
$10x^2 + 2x$
6. $-4t(-t - 5)$
* Multiply $-4t$ by $-t$: Negative times negative is positive. $4 \cdot 1 = 4$, and $t \cdot t = t^2$. Result: $4t^2$
* Multiply $-4t$ by $-5$: Negative times negative is positive. $4 \cdot 5 = 20$, keep the $t$. Result: $+20t$
* Combine them:
$4t^2 + 20t$
7. $-3y(4y - 2)$
* Multiply $-3y$ by $4y$: $-3 \cdot 4 = -12$, and $y \cdot y = y^2$. Result: $-12y^2$
* Multiply $-3y$ by $-2$: Negative times negative is positive. $3 \cdot 2 = 6$, keep the $y$. Result: $+6y$
* Combine them:
$-12y^2 + 6y$
8. $-z(-5z - 6)$
* Multiply $-z$ by $-5z$: Negative times negative is positive. $1 \cdot 5 = 5$, and $z \cdot z = z^2$. Result: $5z^2$
* Multiply $-z$ by $-6$: Negative times negative is positive. $1 \cdot 6 = 6$, keep the $z$. Result: $+6z$
* Combine them:
$5z^2 + 6z$
Final Answer:
$3p(-4p + 2) = -12p^2 + 6p$
$-4a(4a - 5) = -16a^2 + 20a$
$5w(-4w - 3) = -20w^2 - 15w$
$6b(-2b - 5) = -12b^2 - 30b$
$-2x(-5x - 1) = 10x^2 + 2x$
$-4t(-t - 5) = 4t^2 + 20t$
$-3y(4y - 2) = -12y^2 + 6y$
$-z(-5z - 6) = 5z^2 + 6z$
Parent Tip: Review the logic above to help your child master the concept of multiplying polynomials worksheet algebra 2.
