CPM Homework Help : PC Problem 8-76

Solve $x^{3}\left(x + 3\right)^{4} + 2x^{2}\left(x + 3\right)^{3} = 0$ . Hint: Let $a = x + 3$ and factor.

Step 1:

Substitute $a$ for $x + 3$ .

$x^{3}a^{4} + 2x^{2}a^{3} = 0$

Step 2:

Factor.

$x^{2}a^{3}\left(ax + 2\right) = 0$

Step 3:

Set each factor equal to $0$ .

$x^{2}a^{3} = 0$ or $ax + 2 = 0$

Step 4:

Substitute $\left(x + 3\right)$ in each equation and solve.

$x^{2}\left(x + 3\right)^{3} = 0$
$x^{2} = 0$
$x = 0$
$\left(x + 3\right)^{3} = 0$
$x = −3$

$ax + 2 = 0$
$\left(x + 3\right)x + 2 = 0$
$x^{2} + 3x + 2 = 0$
$\left(x + 2\right)\left(x + 1\right) = 0$
$x = −2; x = −1$