Warm up problem

We have $x+y=0$

Hence, $y=-x$

Plugging into the equation given, we get :

$(-x\times x^{4}) + x^{5}=32$

$x^{5} = 32$

Hence, $x= \boxed{2}$

$x + y = 0\\ x = -y\\ \implies yx^4 + 2x^5 = 32\\ -x^5 + 2x^5 = 32\\ x^5 = 32\\ x = \color{#69047E}{\boxed{2}}$

x+y=0 therefore y=-x

((-x) (x^4)) + 2x^5 =32

-x^5 + 2x^5 = 32

x^5=32

x=2

First separate the equation into yx^4+x^5+x^5+x^5, then factor out common terms x^4(x+y)+x^5. Now, we know that x+y=0, so we can negate the first part of the equation and we are left with x^5=32 therefore x=2.