To factorize the quadratic expression [tex]\(5y^2 + 2ay - 3a\)[/tex], let's follow these detailed steps:
1. Write down the original expression:
[tex]\[
5y^2 + 2ay - 3a
\][/tex]
2. Identify the coefficients:
- The coefficient of [tex]\(y^2\)[/tex] is [tex]\(5\)[/tex].
- The coefficient of [tex]\(y\)[/tex] is [tex]\(2a\)[/tex].
- The constant term is [tex]\(-3a\)[/tex].
3. Perform the factorization:
- We are looking to express the quadratic trinomial in the form [tex]\((My + N)(Py + Q)\)[/tex], such that when expanded, it gives us [tex]\(5y^2 + 2ay - 3a\)[/tex].
- Here, [tex]\(M\)[/tex], [tex]\(N\)[/tex], [tex]\(P\)[/tex], and [tex]\(Q\)[/tex] must be chosen so that the product of the outer and inner terms sum up to the middle term [tex]\(2ay\)[/tex] and the product of the [tex]\(MQ\)[/tex] terms should be equal to [tex]\(5y^2\)[/tex] and the [tex]\(NP\)[/tex] terms should be equal to [tex]\(-3a\)[/tex].
4. Determine the factor pairs of the constant term:
- To factor [tex]\(5y^2 + 2ay - 3a\)[/tex], we note that there are no factor pairs that consistently resolve into simpler quadratic or binomial factors.
5. Check if the expression can be factored into simpler polynomials:
- Upon closer inspection and individual attempts to factorize noted quadratics lead us back to the original expression. Hence it appears the expression itself is in the most reduced factorized form.
6. Conclude the result:
- The quadratic polynomial [tex]\(5y^2 + 2ay - 3a\)[/tex] does not factorize further using integer coefficients.
Therefore, the factored version of the given expression remains:
[tex]\[
5y^2 + 2ay - 3a
\][/tex]
Since further simplification or factorization using integer coefficients isn't possible, we conclude that the polynomial is already in its simplest form.
