5x 2 + x + 3 = (x + 3)(5x + .)
To determine what the "." should be replaced with to make the equation true, we can expand the right-hand side expression and compare it to the left-hand side.
Expanding the right-hand side, where the "." represents an unknown constant 'a', we have:
(x + 3)(5x + a) = 5x(x) + 5x(a) + x(3) + a(3)
= 5x^2 + 5ax + 3x + 3a
Now, let's compare this to the left-hand side of the equation:
5x^2 + x + 3
Matching coefficients, we see that:
For the x^2 term: We already have a match, because there is a 5x^2 on both sides.
For the x term: We have 5ax + 3x on the right and x on the left. We need to find 'a' such that:
5a + 3 = 1
We can solve this for 'a':
5a = 1 - 3
5a = -2
a = -2/5
For the constant term: We have 3a on the right and 3 on the left. If a = -2/5, then:
3a = 3(-2/5) = -6/5, which does not match the constant term on the left-hand side (which is 3).
Thus, there's a contradiction: we can't have 3a equal to both -6/5 (from the x term condition) and 3 (to match the constant terms). Since the left-hand side cannot be factored into the form (x + 3)(5x + a) with real coefficients while maintaining equivalence (the linear and constant terms cannot be matched simultaneously), the equation as given seems to be incorrect.
An equation of the form on the left-hand side (5x^2 + x + 3) typically can't be factored into integer or simple fractional factors due to the 5 in front of x^2 and the 3 as the constant, unless it's a perfect trinomial or another special case.
If you still believe the equation can be factored into a form similar to (x + 3)(5x + a), please double-check the original equation or provide any additional constraints that could lead to a solution.