Let $f(x) = x^4-3x^2 + 2$ and $g(x) = 2x^4 - 6x^2 + 2x -1$. Let $a$ be a constant. What is the largest possible degree of $f(x) + a\cdot g(x)$?

To find the largest possible degree of $f(x) + a \cdot g(x)$, we need to determine the degree of the polynomial resulting from this addition.

For polynomials, the degree is determined by the highest power of $x$ with a non-zero coefficient.

First, let's find the degree of $f(x)$ and $g(x)$ separately:

The degree of $f(x)$ is determined by the highest power of $x$ in the polynomial $x^4 - 3x^2 + 2$. The highest power is $x^4$, so the degree of $f(x)$ is $4$.

The degree of $g(x)$ is determined by the highest power of $x$ in the polynomial $2x^4 - 6x^2 + 2x - 1$. The highest power is $x^4$, so the degree of $g(x)$ is also $4$.

Now, when we add $f(x)$ and $a \cdot g(x)$, we simply combine like terms. The resulting polynomial will have the same degree as the highest degree term in the sum.

Let's compute the sum:

$f(x) + a \cdot g(x) = (x^4 - 3x^2 + 2) + a \cdot (2x^4 - 6x^2 + 2x - 1)$.

Expanding and combining like terms, we get:

$f(x) + a \cdot g(x) = (2a + 1) x^4 + (a - 3) x^2 + 2a \cdot x + (2 - a)$.

The highest degree term in this polynomial is $x^4$, and its coefficient is $2a + 1$. Therefore, the largest possible degree of $f(x) + a \cdot g(x)$ is $4$.