This question is inspired by exercises 2.51-2.52 of Eisenbud's "3264".
Here $\mathbb{P}^n$ denotes the $n$-dimensional projective space over an algebraically closed field of (for safety) characteristic not 2 or 3. Let $X\subset\mathbb{P}^9$ be the locus of cubics of the form $2L+M$ for lines $L,M\subset \mathbb{P}^2.$
$X$ can be identified with the image of a map $\mu:\mathbb{P}^2\times \mathbb{P}^2\to\mathbb{P}^9$ which takes a pair of linear forms $F,G$ to $F^2G.$ If $\xi\in A^1(\mathbb{P}^9)$ is the hyperlane class in the Chow ring of $\mathbb{P}^9$, then $\mu^*(\xi)=2\alpha+\beta$ for $\alpha,\beta$ the pullbacks of hyperplane classes of $\mathbb{P}^2$ to the product. $\mu$ is birational, so we find
$$\deg X=\deg [X]\cdot\xi^4=\deg \mu^*(\xi^4)=\deg(2\alpha+\beta)^4=\deg 4{4\choose 2}\alpha^2\beta^2=24.$$
Exercise 2.52 then suggests a different way, which should give the wrong answer, at least without additional considerations. Let's intersect $X$ with 4 general hyperplanes of the form $H_p=\{\text{cubics vanishing at }p\in\mathbb{P}^2\}.$ This should be the same as counting unions of double lines and lines passing through 4 points. There are 6 of them: there are ${4\choose 2}$ ways to choose the pair of points through which to draw a double line and then draw a line through the other 2. The book then asks to show why this result is wrong and does not give the correct degree.
Now this is where I got stuck. Let's compute the tangent space to $X$ at a general point, i.e. at a point where $L$ and $M$ are distinct. The projectivized tangent space there is $$\mathbb{T}_{L^2M}(X)=\mathbb{P}\{C\in\mathbb{P}^9\ |\ C\text{ vanishes at } L\text{ and having a double zero at } L\cap M\}\cong$$
$$\cong\mathbb{P}\{\text{quadratic homogeneous polynomials vanishing at } L\cap M\}.$$
Intersecting with 4 hyperplanes $H_{p_i}$ should then introduce four linear constraints on this space, so the intersection $H_{p_1}\cap\ldots\cap H_{p_4}\cap X$ is transverse, so the degree of $X$ is just the number of points of intersection. Evidently, this reasoning must be wrong somewhere, but I just can't see it.
Any help greatly appeciated, as always!