26) $$ 2 x^{3} y+12 x^{2} y+18 x y $$

Question

UpStudy AI · Accepted Answer

We start with the given expression:

$$
2x^{3}y + 12x^{2}y + 18xy
$$

**Step 1: Factor out the greatest common factor (GCF).**

Each term has a factor of $$2xy$$, so we factor it out:

$$
2x^{3}y + 12x^{2}y + 18xy = 2xy \left( x^{2} + 6x + 9 \right)
$$

**Step 2: Factor the quadratic expression.**

Now, factor the quadratic $$x^{2} + 6x + 9$$. Notice that:

$$
x^{2} + 6x + 9 = \left( x + 3 \right)^2
$$

(this is because $$(x+3)^2 = x^2 + 2\cdot3\cdot x + 3^2 = x^2 + 6x + 9$$).

**Step 3: Write the final answer.**

Substitute the factored form of the quadratic back:

$$
2x^{3}y + 12x^{2}y + 18xy = 2xy \left( x + 3 \right)^2
$$
The expression simplifies to $$2xy(x + 3)^2$$.

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