We are asked to factor the polynomial $x^4 + 7x^2y^2 + 10y^4$.

1. Problem Description

We are asked to factor the polynomial $x^4 + 7x^2y^2 + 10y^4$.

2. Solution Steps

Let $u = x^2$ and $v = y^2$. Then the polynomial becomes $u^2 + 7uv + 10v^2$.

We are looking for two numbers that multiply to 10 and add to

7. The numbers 2 and 5 satisfy this condition.

Therefore, $u^2 + 7uv + 10v^2 = (u + 2v)(u + 5v)$.

Now substitute back $x^2$ for $u$ and $y^2$ for $v$.

$(u + 2v)(u + 5v) = (x^2 + 2y^2)(x^2 + 5y^2)$.

The factored form is $(x^2 + 2y^2)(x^2 + 5y^2)$.

3. Final Answer

$(x^2+2y^2)(x^2+5y^2)$