Kathy Williams

2021-12-11

Express as a polynomial. $\left({r}^{2}+8r-2\right)\left({r}^{2}+3r-1\right)$

### Answer & Explanation

Matthew Rodriguez

Step 1
Given expression:
$\left({r}^{2}+8r-2\right)\left({r}^{2}+3r-1\right)$
Step 2
Consider,
$\left({r}^{2}+8r-2\right)\left({r}^{2}+3r-1\right)={r}^{2}\left({r}^{2}+3r-1\right)+8r\left({r}^{2}+3r-1\right)-2\left({r}^{2}+3r-1\right)$...Use the distributive property.
$=\left[{r}^{2}\left({r}^{2}\right)+{r}^{2}\left(3r\right)+{r}^{2}\left(-2\right)\right]+\left[\left(8r\right)\left({r}^{2}\right)+\left(28r\right)\left(3r\right)+\left(8r\right)\left(-2\right)\right]-\left[2{r}^{2}+\left(2\right)\left(3r\right)-\left(2\right)\left(1\right)\right]$
$={r}^{4}+3{r}^{3}-2{r}^{2}+8{r}^{3}+24{r}^{2}-16r-2{r}^{2}-6r+2$
$={r}^{4}+\left(3{r}^{3}+8{r}^{3}\right)+\left(-2{r}^{2}+24{r}^{2}-2{r}^{2}\right)+\left(-16r-6r\right)+2$...Collect like terms together.
$={r}^{4}+11{r}^{3}+20{r}^{2}-22r+2$
Thus,
$\left({r}^{2}+8r-2\right)\left({r}^{2}+3r-1\right)={r}^{4}+11{r}^{3}+20{r}^{2}-22r+2$

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