Question

match each polynomial function to its graph.
$f(x) = -4x^{4} - 32x^{3} - 88x^{2} - 96x - 36 = -4(x + 3)^{2}(x + 1)^{2}$
$g(x) = -x^{4} - 3x^{2} - 2 = -(x^{2} + 2)(x^{2} + 1)$
$f(x) = -4x^{4} - 32x^{3} - 88x^{2} - 96x - 36$ $g(x) = -x^{4} - 3x^{2} - 2$

Explanation:

Step1: Analyze zeros of $f(x)$

$f(x) = -4(x + 3)^2(x + 1)^2$, so zeros at $x=-3$ and $x=-1$ (double roots). The left graph touches the x-axis at these points, consistent with double roots.

Step2: Analyze zeros of $g(x)$

$g(x) = -(x^2 + 2)(x^2 + 1)$, which has no real zeros (since $x^2 + 2 > 0$ and $x^2 + 1 > 0$ for all real $x$). The right graph does not cross/touch the x-axis, consistent with no real zeros.

Step3: Confirm end behavior

Both are even-degree polynomials with negative leading coefficients, so end behavior is down on both ends, matching both graphs. The zero analysis distinguishes them.

Answer:

$f(x) = -4x^4 - 32x^3 - 88x^2 - 96x - 36$ matches the left graph; $g(x) = -x^4 - 3x^2 - 2$ matches the right graph.