Question

If x+a is a factor of 2x²+2ax+5x+10,find a.

Answer 1

If $x+a$ is a factor of the polynomial, it means that $-a$ is a root of the polynomial.

In fact, if a polynomial $p(x)$ has a root $x_0$ (i.e. $p(x_0)=0$), then the polynomial is divisible by $(x-x_0)$

In your case, you know that the polynomial is divisible by $(x+a)=(x-(-a))$, so $-a$ is a solution.

Let's evaluate $p(-a)$ and set it to zero:

$2(-a)^2+2a(-a)+5(-a)+10 = 0 \iff 2a^2-2a^2-5a+10=0 \iff -5a+10 = 0 \iff a = 2$

We can check the answer: if $a=2$ the polynomial becomes

$2x^2+2(2)x+5x+10 = 2x^2+9x+10$

And we're claiming that $-a=-2$ is a root of this polynomial:

$2(-2)^2+9(-2)+10 = 2\cdot 4 - 18 + 10 = 0$